LB 
H3 


GIFT  OF 


National  Education  Association 


Final  Report 

OF  THE 

National  Committee  of  Fifteen 


ON 


Geometry  Syllabus 


July,  19 1 2 


/ 


\   O  I    VC      J     ^ 


K'3 


Composed  and  Printed  By 

The  University  of  Chicago  Press 

Chicago,  Illinois.  U.S.A. 


is- 


FINAL  REPORT  OF  THE  NATIONAL  COMMITTEE 
OF  FIFTEEN  ON  GEOMETRY  SYLLABUS 


SECTION  A.    HISTORICAL  INTRODUCTION 

The  committee  regards  the  historical  statement  printed  in  the  volume 
of  Proceedings  for  191 1,  pp.  607-35  inclusive,  and  constituting  Section  A 
of  the  final  report,  as  most  important  and  calls  attention  to  the  following 
additional  reference  to  this  section: 

The  committee  recommends  the  foregoing  historical  sketch  to  the 
careful  consideration  of  teachers  of  geometry.  Special  attention  is  called 
to  the  age-long  contest  between  the  extreme  formalists  and  the  extreme 
utilitarians.  The  committee  stands  for  neither  extreme  position.  It 
recommends  a  reasonable  attention  to  exercises  in  concrete  setting,  such, 
for  instance,  as  simple  problems  involving  the  trigonometric  ratios  in 
connection  with  similar  triangles,  or  such  applications  as  those  shown  on 
pages  21-24  of  this  report.  But  in  so  doing  it  does  not  recommend  dimin- 
ishing attention  to  the  logical  side  of  the  subject,  but  rather  a  quickening 
of  the  logical  sense  thru  a  more  rational  distribution  of  emphasis  which  will 
make  for  economy  of  both  time  and  mental  energy  in  mastering  standard 
theorems  and  leave  opportunity  for  a  broader  view  of  the  subject  in  its 
concrete  relations.    See  Sections  D  and  E. 

SECTION  B.    LOGICAL  CONSIDERATIONS 
AXIOMS 

(a)  Nomenclature. — ^The  best  historical  usage  distinguishes  between 
Axioms  (Euclid's  "Common  Notions'')  and  Postulates  (Euclid's  aitemata 
or  "requests")  by  including  in  the  former  certain  general  statements 
assumed  for  all  mathematics,  and  in  the  latter  certain  specifically  geometric 
concessions.  These  names  and  this  distinction  are  now  in  general  use  and 
there  seems  no  good  reason  for  attempting  to  change  them.  However, 
teachers  who  may  wish  to  use  the  single  term  "assumptions"  to  cover  both, 
or  to  use  the  term  "axiom"  to  mean  any  proposition  whose  truth  is  postu- 
lated, thus  making  axiom  and  postulate  synonymous,  should  be  free  to 
do  so. 

(b)  General  Nature. — It  is  evident  that  strict  mathematical  science 
would  lead  us  to  seek  and  to  recommend  an  "irreducible  minimimi"  of 
assumptions,  while  educational  science  leads  us  to  see  that  such  a  list  would 
be  unintelligible  to  pupils  and  therefore  unusable  in  the  schools.    Since  we 

3 


NATIONAL  EDUCATION  ASSOCIATION 


cannot  recommend  the  adoption  of  a  set  of  assumptions  along  the  Hilbert 
line,  we  therefore  lay  down  the  general  line  of  axioms  and  postulates 
needed  in  geometry,  without  insisting  upon  an  exact  list  or  upon  any 
particular  phraseology. 

(c)  General  List  of  Axioms. — 

As  to  the  nature  of  the  quantities,  positive  quantities  are  to  be  under- 
stood. When  the  negative  quantity  enters  into  elementary  geometry  it  is 
in  the  discussion  of  propositions  and  not  in  cases  in  which  the  axioms  are 
directly  employed.  For  example,  it  is  desirable  not  to  confuse  beginners 
in  geometry  by  the  question  of  dividing  unequals  by  negative  numbers. 

Operations  upon  equal  quantities. — It  should  be  stated,  preferably  in  a 
series  of  axioms,  that  if  equals  are  operated  upon  by  equals  in  the  same  way, 
the  results  are  equal:  i.e.,  if  a  =  b  and  x=y,  then  a-\-x  =  h-\-y,  a—x  =  b—y 
(where  a>x),  ax=by,  etc. 

Operations  upon  unequal  quantities. — It  should  be  stated  that  if  unequals 
are  operated  on  by  equals  in  the  same  way,  the  results  are  unequal  in  the 
same  order;  i.e.,  if  a>b  and  x=yy  then  a-\-x>b-\-y,  etc.  These  various 
cases  enter  into  elementary  geometry,  and  this  assumption  should  be  stated 
in  such  a  manner  that  the  student  can  easily  refer  to  it  in  his  work. 

There  is  also  the  assumption  that  if  unequals  are  added  to  unequals  in 
the  same  order,  the  sums  are  unequal  in  the  same  order,  and  that  if  unequals 
are  subtracted  from  equals  the  remainders  are  unequal  in  the  reverse  order, 
these  being  the  only  ones  relating  to  inequalities  that  are  needed  in  elemen- 
tary geometry. 

As  to  substitutions. — In  geometry  it  is  continually  necessary  to  make  use 
of  the  assumption  that  a  quantity  may  be  substituted  for  its  equal  in  an 
equation  or  in  an  inequality.  Often  this  assumes  the  common  form  that 
"quantities  that  are  equal  to  the  same  quantity  are  equal  to  each  other.'* 
The  committee  recommends  this  axiom. 

Inequality  among  three  quantities. — It  is  necessary  to  say  in  geometry 
that  ii  a>b  and  b>c  then  a>c,  and  an  axiom  to  this  effect  is  necessary. 

The  whole  and  its  part. — ^Altho  the  definition  of  "whole"  might  be  given 
in  such  a  manner  as  to  render  unnecessary  the  usual  axiom,  it  seems 
advisable  to  make  the  statement  in  the  ordinary 'form. 

It  is  to  be  understood  that  in  applying  these  axioms  to  geometric 
magnitudes,  the  letters  used  refer  to  the  numerical  measures  of  such  mag- 
nitudes, and  as  such  belong  to  arithmetic  and  algebra,  thus  giving  the 
theoretical  basis  for  the  correlation  of  geometry  with  these  subjects. 

Summary. — Axioms  covering  the  above  points  are  of  advantage  in  the 
practical  teaching  of  geometry,  but  this  committee  has  no  recommendation 
to  make  as  to  order  or  phraseology     They  may  be  summarized  as  follows: 

If  a=6  and  ii;=y,  then  '* ' 

(i)  a-\-x=b-\-y.  (3)  ax^by. 

(2)  a—x=b-'y  {(a>x).  (4)  a/x^b/y. 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS 


(5)  a"  =  ^>'*  and  ya=  \/h,  where  »  is  a  positive  integer. 

(6)  If  a>b  and  x=y,  then  a-\-x>h-\-y,  ax>by,  a/x>b/y,  and  if 
a>x,  then  a^x>b—y. 

(7)  If  a>b  and  c>d,  then  a+c>^+^,  and  if  x  =  y,  then  a;— fl<>'— ^>. 

(8)  If  a=iK;,  and  6  =  x,  then  a  =  b. 

(9)  If  ic  =  a,  we  may  substitute  a  for  jc  in  an  equation  or  in  an  inequality. 

(10)  If  a>b  and  if  b>c,  then  a>c. 

(11)  The  whole  is  greater  than  any  of  its  parts,  and  is  equal  to  the  sum 
of  all  its  parts. 

(d)  General  List  of  Postulates. — 

(i)  One  straight  line  and  only  one  can  be  drawn  thru  two  given  points. 

Corollary  i.     Two  points  determine  a  straight  line. 

Corollary  2.     Two  straight  lines  can  intersect  in  only  one  point, 

(2)  A  straight  line-segment  may  be  produced  to  any  required  length. 

This  includes  one  postulate  and  one  problem  of  Euclid,  and  so  mani- 
festly depends  upon  the  simplest  uses  of  straight  edge  and  compasses  as  to 
be  a  proper  geometric  assumption. 

(3-)  A  straight  line  is  the  shortest  line  between  two  points. 

(4)  A  circle  may  be  described  with  any  given  point  as  a  center  and  any  given 
line-segment  as  a  radius. 

(5)  Any  figure  may  be  moved  from  one  place  to  another,  without  altering 
its  size  or  shape. 

(6)  All  straight  angles  are  equal. 

This  and  the  following  corollaries  may  be  included  among  the  theorems 
for  informal  proof  under  Section  E. 

Corollary  1,  All  right  angles  are  equal. 

Corollary  2.  From  a  point  in  a  line  only  one  perpendicular  can  be  drawn 
to  the  line. 

Corollary  3.  Equal  angles  have  equal  complements ,  equal  supplements ^ 
and  equal  conjugates. 

Corollary  4.  The  greater  of  two  angles  has  the  less  complement,  the  less 
supplement,  and  the  less  conjugate. 

The  above  axioms  and  postulates  may  be  recomm.ended  for  use  as  soon 
as  the  formal  proof  of  propositions  is  begun,  the  postulate  of  parallels  being 
introduced  when  needed,  as  follows: 

Postulate  of  Parallels.  Thru  a  given  point  one  line  and  only  one  can  be 
drawn  parallel  to  a  given  line. 

The  question  of  limits  is  considered  later.  It  is  not  deemed  desirable  to 
postulate  explicitly  the  existence  of  such  concepts  as  point,  line,  and  angle, 
nor  to  assume  that  a  line  drawn  thru  a  point  in  a  triangle  must  cut  the 
perimeter  twice,  nor  to  add  a  postulate  of  continuity.  It  is  well,  how- 
ever, for  teachers  to  mention  thai  .  _h  assumptions  are  always  tacitly 
made. 

In  any  case  the  committee  feels  that  a  certain  amount  of  care  should  be 


NATIONAL  EDUCATION  ASSOCIATION 


taken  in  fixing  the  location  of  points  and  lines  and  proving  that  lines  inter- 
sect, when  the  accuracy  of  the  proof  in  question  might  be  affected  by 
ignoring  such  details. 

DEFINITIONS 

(a)  New  Terms. — 

(i)  General  principle. — It  is  unwise  for  individual  teachers  or  writers  to 
introduce  terms  beyond  those  actually  in  common  use  in  geometry,  or  to 
change  the  accepted  meaning  of  common  terms,  unless  there  seems  to  be  a 
very  definite  advantage  in  the  new  term  and  an  unquestionable  sanction  in 
the  mathematical  world.  In  particular,  the  substitution  of  a  new  term  for 
an  old  one  to  denote  the  same  concept  is  undesirable. 

(2)  Type  of  terms  that  may  safely  he  added  to  those  ot  the  older  elementary 
geometry. — Congruent,  because  this  is  so  widely  used  both  here  and  abroad, 
and  because  it  avoids  the  loose  use  of  equal  and  the  long  forms  of  identically 
equal,  and  equal  in  all  their  parts. 

(3)  Types  of  terms  that  may  safely  he  dropped. — Scholium,  because  this 
has  been  so  generally  abandoned,  and  because  it  is  unnecessary;  mixed  line, 
an  antiquated  term  of  no  value  in  elementary  geometry.  Other  suchterms 
are  trapezium  and  rhomboid. 

(4)  Types  of  terms  that  seem  of  too  douhtful  advantage  to  be  recommended 
definitely  by  this  committee,  teachers  being  left  free  to  use  them  if  they 
desire,  the  terms  thus  being  given  an  opportunity  to  make  their  way  if  they 
possess  real  merit. — Ray,  a  term  that  has  abundant  sanction  in  higher 
geometry,  but  may  be  dispensed  with  in  elementary  work.  Other  such 
terms  are  mid-join  (for  median),  cuboid  (for  rectangular  parallelepiped), 
n-gon  (for  polygon  of  n  sides),  and  sect  (for  segment  of  a  straight  line). 

(5)  Types  of  terms  that  are  used  with  a  different  meaning  in  higher 
geometry,  and  that  may  properly  be  used  in  elementary  geometry  with  the 
more  recent  signification. — Circle  as  meaning  the  line,  which  is,  indeed,  the 
primitive  Greek  meaning;  circumference,  as  meaning  the  length  of  the  circle 
— these  usages  requiring  a  re-defining  of  segment  of  circle,  semicircle,  area  of 
circle,  and  other  obvious  terms. 

The  committee  recognizes  also  the  tendency  to  unify  the  usage  of  such 
terms  as  polygon  and  sphere  in  elementary  and  higher  geometry.  This 
tendency  should  be  encouraged.  In  any  case,  it  is  essential  that  the  pupil 
should  understand  clearly  what  the  terms  mean  in  the  statements  and 
proofs  of  the  propositions. 

(b)  Symbols.— 

(i)  General  principle. — No  symbols  should  be  recommended  beyond 
such  as  are  already  in  wide  use  in  elementary  geometry,  and  any  that  are 
unnecessary  or  are  not  generally  accepted  should  be  abandoned.  The 
elaboration  of  personal  symbolism,  sometimes  to  the  point  of  eccentricity, 
is  such  as  to  be  cumbersome  in  the  mathematics  of  the  present. 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS 


(2)  Recommendations. — The  committee  feels  that  the  common  symbols 
of  algebra,  most  of  which  are  known  to  the  pupil  beginning  geometry,  and 
such  obvious  symbols  as  those  for  perpendicular,  triangle,  circle,  square, 
and  parallel,  are  all  that  are  needed  in  a  course  in  elementary  geometry,  and 
that  it  is  unnecessary  to  specify  these  symbols  in  detail.  It  appears  that 
there  is  no  generally  received  symbol  for  congruence,  the  symbols  =,  =, 
and  ^  all  being  in  use,  and  it  seems  best  to  recognize  this  fact,  leaving 
teachers  at  present  to  decide  the  question  for  themselves.  In  due  time  a 
general  consensus  of  opinion  may  lead  to  some  definite  usage,  and  it  is  the 
feeling  of  the  committee  that  the  second  symbol  given  is  a  desirable 
one. 

(c)  Distribution  of  Definitions. — ^The  committee  recommends  that  new 
terms  be  taught  when  the  time  arrives  for  using  them.  This  allows  a 
teacher  to  use  a  book  in  which  the  definitions  are  massed  or  one  in  which 
they  are  scattered,  but  it  encourages  teaching  them  on  the  latter  plan.  It 
is  recognized  that  the  massed  plan  has  the  advantage  of  a  dictionary 
arrangement,  and  this  is  a  plan  that  a  textbook  writer  might  reasonably 
adopt,  but  it  is  not  a  plan  to  be  followed  in  the  actual  teaching  of  the 
terms. 

(d)  The  Defined  and  the  Undefined. — The  attention  of  teachers  is 
called  to  the  fact,  now  coming  to  be  well  recognized,  that  certain  terms  in 
geometry  must  be  looked  upon  as  undefined. 

Certain  concepts  are  so  elementary  that  no  simpler  terms  exist  by  which 
to  define  them,  altho  they  can  easily  be  explained.  For  example,  poini,  linCy 
surface,  space,  angle,  straight  line,  curve.  The  committee  recommends  that 
teachers  give  more  attention  to  instilling  a  clear  concept  of  such  terms  and 
none  to  exact  definition. 

On  the  other  hand,  the  committee  recommends  the  careful  definition  of 
readily  definable  terms,  where  these  definitions  are  parts  of  subsequent 
proofs,  such  definitions  to  be  memorized  exactly  or  in  their  essentials.  For 
example,  right  angle,  square,  isosceles  triangle,  parallelogram. 

There  is  a  further  class  of  easily  defined  terms,  where  the  definition  is 
not  made  the  basis  of  a  proof,  and  it  seems  obvious  to  the  committee  that 
the  memorizing  of  the  exact  wording  of  such  definitions  is  not  a  wise  expendi- 
ture of  time.  Such  terms  are  hexagon,  heptagon,  reentrant  angle,  concave 
polygon,  etc. 

(e)  The  Form  of  Definition. — A  definition  may  begin  with  the  term 
defined,  as  in  a  dictionary;  or  it  may  close  with  the  word  defined;  or  it  may 
at  times  contain  the  word  in  the  midst  of  the  sentence.  The  committee 
feels  that  it  is  of  no  moment  which  of  these  forms  is  taken,  or  that  the 
definition  be  embodied  in  a  single  sentence.  A  definition  that  is  to  be 
memorized  as  the  basis  of  a  proof  should  be  as  nearly  scientific  as  the  powers 
of  a  beginner  in  geometry  will  justify,  containing  only  terms  that  are 


8  NATIONAL  EDUCATION  ASSOCIATION 

simpler  than  the  term  defined,  not  being  tautological,  and  being  reversible 
— but  further  than  this  it  seems  unwise  to  attempt  to  specify  the  form  of  a 
definition. 

INFORMAL  PROOFS 

(a)  Justification. — It  is  not  pretended  that  elementary  geometry  is 
a  perfect  piece  of  logic.  In  general,  the  modern  departures  from  Euclid 
have  sacrificed  logic  for  other  ends,  and  even  Euclid's  Elements  was  not 
without  numerous  logical  imperfections.  That  is  to  say,  it  has  always  been 
considered  justifiable  to  sacrifice  logic  to  a  greater  or  less  degree.  The 
principle  is  that  a  logical  sequence  should  be  maintained,  and  formal  proofs 
of  propositions  necessary  to  the  sequence  should  be  required,  so  far  as  this 
is  consonant  with  the  educational  principle  of  adapting  the  matter  to  the 
mind  of  the  learner.  Now  in  many  cases  it  happens  that  an  informal  and 
confessedly  incomplete  proof  is  more  convincing  to  a  beginner  than  a  formal 
and  complete  one,  and  is  less  discouraging  because  it  postpones  the  minor 
and  seemingly  unimportant  steps  to  a  time  when  their  importance  may  be 
appreciated  and  the  proofs  understood. 

(b)  Types. — ^To  be  specific,  the  following  are  types  of  propositions 
that  are  better  passed  over  by  the  beginner  without  a  formal  statement, 
being  introduced  at  the  proper  points  in  the  development,  or  with 
informal  proof,  than  proved  in  the  Euclidean  fashion: 

//  one  straight  line  meets  another  the  sum  of  the  two  adjacent  angles  is  a 
straight  angle,  and  conversely  (and  related  propositions) ; 

All  straight  angles  are  eqical  (a  proper  postulate  with  related  corollaries); 

Two  straight  lines  can  intersect  in  only  one  point; 

A  straight  line  can  have  hut  one  point  of  bisection  (and  the  related  case 
for  angles) ; 

The  bisectors  of  vertical  angles  lie  in  one  straight  line; 

Polygons  similar  to  the  same  polygon  are  similar  to  each  other; 

If  one  angle  is  greater  than  another,  its  complement  is  less  than  the  com- 
plement of  the  other  (and  related  propositions) ; 

A  straight  line  can  cut  a  circle  {circumference)  in  two  points  only; 

Circles  of  equal  radii  are  equal  (and  related  statements) ; 

All  radii  of  the  same  circle  are  equal  (and  similarly  for  diameters); 

A  circle  can  have  but  one  center; 

And  propositions  relating  to  the  conditions  under  which  two  circles 
(circumferences)  intersect. 

It  should  be  understood  that  these  propositions  are  merely  types,  and 
that  others  of  the  same  type  may  be  treated  in  the  same  way,  as  specified 
in  Section  E  of  this  report. 

(c)  Experience  of  Other  Countries. — It  is  the  experience  of  all  countries 
where  Euclid  is  not  taught  that  good  results  follow  from  the  use  of  a  reason- 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS 


able  number  of  such  informal  proofs.  The  German  and  Austrian  textbooks 
are  especially  given  to  such  procedure,  and  the  results  seem  to  have  teen 
favorable  rather  than  otherwise.  The  number  of  propositions  formally 
proved  in  a  German  textbook  is  notably  less,  for  example,  than  in  a  cor- 
responding French  textbook. 

(d)  Dangers. — It  is  evident,  however,  that  we  may  easily  go  to  a 
dangerous  and  ridiculous  extreme  in  this  matter.  With  all  of  the  experi- 
ments at  improving  Euclid  the  world  has  really  accomplished  very  little 
except  as  to  the  phraseology  of  propositions  and  proofs;  the  standard 
propositions  remain,  and  if  geometry  has  any  justification,  apart  from  its 
kindergarten  aspect  (which  requires  but  a  short  time),  most  of  these  proposi- 
tions will  continue  to  be  proved,  and  should  continue  to  be  proved.  These 
propositions,  whether  in  the  Euclid  or  Legendre  arrangement,  number  in 
the  neighborhood  of  i6o  for  plane  geometry.  Of  this  number  upward  of 
one  hundred  must  receive  formal  proof  in  any  well-regulated  course  in 
geometry. 

TREATMENT  OF  LIMITS  AND  INCOMMENSURABLES 

(a)  Present  Status. — It  is  generally  agreed  that  the  present  treatment 
of  this  subject  is  open  to  two  objections:  (i)  it  is  not  sufficiently  understood 
by  the  student  to  make  it  worth  the  while,  and  (2)  it  is  not  scientifically 
sound. 

(b)  Remedies  Proposed. — Corresponding  to  the  two  defects  mentioned 
two  remedies  have  been  proposed:  (i)  to  make  it  less  formal  and  technical, 
so  that  it  shall  be  better  understood,  and  (2)  to  abandon  the  incommensur- 

•able  case  altogether  in  secondary  education. 

(c)  The  Position  of  This  Committee. — This  committee  recommends 
that  in  elementary  geometry  the  nature  of  incommensurables  and  limits 
be  explained,  but  that  the  subject  no  longer  be  required  for  entrance  to 
college  or  be  included  in  official  examinations.  It  recommends  that  the 
schools  treat  the  subject  as  fully  beyond  this  point  as  circumstances  seem 
to  demand,  and  to  this  end  reference  is  made  to  the  syllabus  given  in 
Section  E. 

The  prime  object  is  to  relieve  the  schools  of  the  necessity  of  teaching 
the  subject,  while  leaving  them  free  to  do  so  if  they  wish. 

TIME  AND  PLACE  IN  THE  CURRICULUM 

(a)  Conventionally. — At  present,  in  America,  plane  geometry  is 
generally  taught  in  the  tenth  school  year  (not  counting  the  kindergarten). 

In  the  East  it  is  completed  in  the  eleventh  school  year,  and  in  the  West 
solid  geometry  is  completed  in  the  eleventh  or  twelfth  year.  In  spite  of 
all  the  discussion  about  constructive  geometry  (intuitive,  metrical,  etc.)  in 
the  first  eight  grades,  carried  on  in  the  past  half-century,  no  generally 


lO  NATIONAL  EDUCATION  ASSOCIATION 

accepted  plan  has  been  developed  to  replace  the  old  custom  of  teaching  the 
most  necessary  facts  of  mensuration  in  connection  with  arithmetic.  We 
have,  therefore,  at  this  time,  algebra  in  the  ninth  school  year,  plane  geome- 
try in  the  tenth,  and  algebra  and  geometry  in  the  eleventh  and  sometimes 
in  the  twelfth. 

(b)  Changes  Suggested. — Certain  changes  in  this  conventional  plan 
have  been  suggested. 

(i)  To  provide  for  preliminary  (inductive,  constructive,  observational) 
work  in  geometry  in  the  elementary  grades.  This  topic  is  discussed  in 
Section  C  of  this  report. 

(2)  To  precede  the  work  in  plane  geometry  by  some  definite  work  in 
geometric  drawing.  Attention  may  be  called  to  the  fact  that  the  recent 
great  advance  in  art  education  has  had  one  disadvantage  from  the  stand- 
point of  geometry,  in  that  geometric  drawing  has  been  abandoned,  and 
that  therefore  some  little  work  in  handling  compasses  and  ruler  must  now 
form  part  of  the  first  steps  in  this  subject. 

(3)  To  unite  geometry  and  algebra,  or  geometry  and  trigonometry. 
This  committee  does  not  feel  that  the  experiments  along  this  line,  which 
have  been  made  in  only  a  few  schools,  have  been  sufiicient  to  determine 
whether  or  not  geometry  should  run  parallel  with  algebi*  in  the  ninth, 
tenth,  and  eleventh  school  years. 

(c)  Position  of  This  Committee. — This  committee  recommends  that 
plane  geometry  be  assigned  not  less  than  one  year  nor  more  than  one  and 
one-half  years  in  the  curriculum,  being  preceded  by  at  least  one  year  of 
algebra  except  where  the  individual  teacher  desires  to  carry  it  along  with 
algebra. 

It  should  be  distinctly  understood  that  owing  to  the  condition  of  unrest 
in  the  entire  field  of  secondary  education  it  is  at  present  impossible  to  give 
any  final  advice  along  any  of  these  lines  of  change.  It  is  probable  that 
many  of  the  readjustments  now  under  general  discussion  will  influence 
every  high-school  curriculum  in  the  course  of  time.  It  is  also  possible  that 
some  of  the  proposed  changes  will  be  adapted  by  the  different  types  of 
secondary  schools  to  their  own  needs,  and  that  they  will  receive  greatly 
varying  emphasis  in  different  localities.  A  certain  amount  of  experimen- 
tation will  undoubtedly  be  necessary  to  test  the  feasibility  of  some  of  the 
proposed  plans.  Great  care  should  be  taken  to  make  all  such  experimenta- 
tion with  due  regard  for  all  that  was  good  in  the  past,  so  that  the  new 
curricula  may  be  the  result  of  evolution  and  not  of  revolution. 

The  most  noteworthy  tendency  in  secondary  education  is  the  desire 
for  more  organic  teaching  and  hence  the  desire  for  more  time.  This 
tendency  finds  its  most  significant  expression  in  the  movement  toward  a 
six-year  curriculum.  It  is  undoubtedly  true  that  in  a  six-year  curriculum 
many  of  the  problems  of  correlation  would  be  brought  nearer  to  a  solution, 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS  II 

that  many  diflficulties  arising,  from  the  present  tandem  system  would 
disappear,  and  that  mathematics  would  be  given  a  place  in  the  curriculum 
more  nearly  commensurate  with  its  importance. 

For  a  brief  account  of  the  six-year  curriculum  the  reader  is  referred  to 
the  book  of  Hanus  entitled  A  Modern  Schooly  published  by  the  Macmillan 
Company,  and  also  to  the  Proceedings  of  the  National  Education  Association 
for  1908. 

PURPOSE   IN   THE   STUDY   OF   GEOMETRY 

(a)  Historical  Review. — Geometry  was  originally,  as  its  name  indicates, 
purely  a  practical  subject.  This  phase  of  its  history  remains  in  the  work 
in  mensuration  in  arithmetic  today.  It  then  became  a  philosophical 
subject,  connecting  with  mysticism  in  the  Pythagorean  school,  being  put 
upon  a  more  solid  scientific  basis  by  the  Platonists,  and  being  crystallized 
by  Euclid  about  300  B.C.  Since  that  time  the  formal  side  has  dominated. 
But  this  formal  side  has  been  attacked  time  after  time,  by  the  astrologers 
and  mystics,  by  the  cathedral  builders  of  the  Middle  Ages,  strongly  by  the 
French  writers  of  the  seventeenth  and  eighteenth  centuries,  recently  by  an 
extreme  school  in  England,  and  at  present  in  a  less  formidable  fashion  in 
our  own  country.  The  results  of  these  attacks  in  so  far  as  they  have 
meant  the  abandoning  of  formal  proofs  have  been  futile. 

(b)  The  Practical  Side. — In  the  high  school  geometry  has  long  been 
taught  because  of  its  mind-training  value  only.  This  exclusive  attention 
to  the  disciplinary  side  may  be  fascinating  to  mature  minds,  but  in  the  case 
of  young  pupils  it  may  lead  to  a  dull  formalism  which  is  unfortunate.  On 
the  other  hand,  those  who  are  advocating  only  a  nominal  amount  of  formal 
proof,  devoting  their  time  chiefly  to  industrial  applications,  are  even  more 
at  fault.  The  committee  feels  that  a  judicious  fusion  of  theoretical  and 
applied  work,  a  fusion  dictated  by  common-sense  and  free  from  radicalism 
in  either  direction,  is  necessary. 

As  to  the  nature  of  the  applications,  the  committee  feels  that  there 
are  several  types  of  genuine  problems,  but  that  many  of  the  so-called  real 
applications  either  are  too  technical  to  be  within  the  grasp  of  the  young 
beginner,  or  represent  methods  of  procedure  that  would  not  be  followed  in 
real  life.  Moreover,  it  should  be  remembered  that  the  very  limited  time 
devoted  to  plane  geometry  (usually  a  single  year)  renders  it  impracticable 
to  introduce  many  of  the  applications  that  might  be  desirable  if  the  time 
were  not  so  restricted. 

(c)  The  Formal  Side. — No  reference  to  the  applications  of  geometry  is 
to  be  construed  to  mean  that  the  committee  feels  that  the  formal  side  should 
suffer,  or  that  geometry  is  wanting  in  a  distinct  disciplinary  value.  A 
formal  treatment  of  geometry,  to  about  the  traditional  extent,  is  necessary 
purely  as  a  prerequisite  to  the  study  of  more  advanced  mathematics,  and 


12  NATIONAL  EDUCATION  ASSOCIATION 

still  more  because  such  treatment  has  a  genuine  culture  value,  for  example, 
in  assisting  to  form  correct  habits  in  the  use  of  English. 

Certain  writers  on  education  have  claimed  that  geometry  has  no 
distinctive  disciplinary  value,,  or  that  the  formal  side  is  so  intangible  that 
algebra  and  geometry  should  be  fused  into  a  single  subject  (not  merely 
taught  parallel  to  each  other),  which  subject  should  occupy  a  single  year 
and  be  purely  utilitarian.  These  writers  fail  to  recognize  the  fundamen- 
tal significance  of  mathematics  in  either  its  intellectual  or  its  material 
bearing. 

(d)  Claims  for  Geometry. — ^Among  the  claims  in  behalf  of  geometry 
the  committee  would  emphasize  the  following: 

Geometry  is  taught  because  of  the  pleasure  it  gives  when  properly 
presented  to  the  average  mind. 

Geometry  is  taught  because  of  the  profit  it  gives  when  properly  pre- 
sented.   For  example: 

(i)  It  is  an  exercise  in  logic,  and  in  types  of  logic  not  generally  met  in 
other  subjects  of  the  school  course,  and  yet  types  which  occur  in  geometry 
in  unusually  simple  setting  and  which  are  easily  carried  over  into  the  actual 
affairs  of  life.  Closely  connected  with  the  logical  element  is  the  training  in 
accurate  and  precise  thought  and  expression  and  the  mental  experience 
and  contact  with  exact  truth. 

This  logic  may  be  no  more  practical  than  literature  or  art  or  any  other 
great  branch  of  learning,  but  its  general  effect  on  the  human  mind  has  been 
doubted  by  such  a  small  number  of  scholars  as  to  render  it  worthy  of  the 
highest  confidence. 

(2)  The  study  of  geometry  leads  also  to  an  appreciation  of  the  depend- 
ence of  one  geometric  magnitude  upon  another,  in  other  words  to  the 
tangible  concept  oi  functionality. 

(3)  The  study  of  geometry  cultivates  space  intuition  and  an  appreciation 
of,  and  control  over,  forms  existing  in  the  material  world,  which  can  be 
secured  from  no  other  topic  in  the  high-school  curriculum. 

(4)  The  value  of  the  applications  of  geometry  to  mensuration  and  the 
satisfaction  derived  by  the  pupil  in  verifying  the  formulas  of  mensuration 
already  met  by  him  in  arithmetic  are  well  recognized  by  all  teachers. 

If  we  had  to  justify  the  position  of  any  other  subject  in  the  curriculum, 
history,  rhetoric,  geography,  biology,  etc.,  it  is  doubtful  whether  equally 
specific  and  cogent  reasons  could  be  found.  If  we  were  to  dismiss  geom- 
etry with  a  few  practical  lessons,  much  more  should  we  be  compelled  to 
dismiss  most  other  subjects  in  the  curriculum  with  the  same  treatment. 

HISTORICAL  NOTES 

Of  the  stimulating  effect  of  occasional  bits  of  historical  information 
given  by  the  teacher  or  the  textbook  there  can  be  no  question.    There  is 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS  13 

plenty  of  material  to  be  found  in  the  well-known  elementary  histories  of 
the  subject.  The  discoverers  of  particular  propositions  are  known  in  a  few 
cases,  and  the  general  story  of  the  subject,  told  informally  as  the  pupil 
proceeds  in  his  study,  adds  a  human  interest  that  is  valuable.  Portraits 
of  famous  mathematicians  may  be  recommended  for  the  schoolroom. 

POINTS   RELATING   TO    SOLID   GEOMETRY 

(a)  Axioms  and  Postulates. — The  list  of  axioms  already  given  need  not 
be  increased,  but  the  following  postulates  may  be  added: 

(i)  One  plane  and  only  one  can  be  passed  thru  two  intersecting  straight  lines. 

Corollary.  A  plane  is  determined  by  three  points  not  in  the  same 
straight  line,  by  a  straight  line  and  a  point  not  in  it,  or  by  two  parallel  lines. 

This  postulate,  which  is  the  analogue  of  the  first  postulate  in  plane 
geometry,  may  also  be  given  as  a  theorem  for  informal  proof. 

(2)  Two  intersecting  planes  have  at  least  two  points  in  common. 

(3)  ^  sphere  may  be  described  with  any  given  point  as  center  and  any  given 
line-segment  as  radius. 

It  is  tacitly  assumed  that  the  figures  described  in  the  course  in  solid 
geometry  exist  and  can  be  made  the  subject  of  investigation;  e.g.,  the  prism, 
pyramid,  cylinder,  cone,  etc. 

It  may  also  be  assumed,  tacitly  or  explicitly,  that  the  various  closed 
solids  have  definite  areas  and  volumes;  e.g.,  that  a  sphere  has  a  definite 
volume  which  is  less  than  that  of  any  circumscribed  convex  polyhedron  and 
greater  than  that  of  any  inscribed  convex  polyhedron. 

(b)  Definitions. — ^Latitude  is  left  to  the  teacher  in  regard  to  the  use  of 
such  terms  as  prismatic  space,  cylindrical  space,  nappes  of  a  cone,  and  some 
of  the  names  suggested  for  a  rectangular  parallelopiped,  which  are  con- 
venient but  not  necessary  in  an  elementary  course. 

After  the  analogy  of  the  circle  defined  as  a  line,  it  is  proper  that  the 
sphere  be  defined  as  a  surface  but  the  more  common  definition  may  be 
retained  if  desired. 

(c)  Purpose. — In  solid  geometry  the  utilitarian  features  play  an  increas- 
ingly important  part.  The  mensuration  involved  in  plane  geometry  is  so 
simple  as  to  be  fairly  well  understood  as  presented  in  arithmetic.  Solid 
geometry,  however,  offers  a  rather  extended  field  for  practical  mensuration 
in  connection  with  algebraic  formulas.  The  subject  is  therefore  particularly 
valuable  for  high-school  classes.  A  further  application  is  found  in  the  power 
afforded  to  visualize  solid  forms  from  flat  drawings,  a  power  that  is  essential 
to  the  artisan  and  valuable  to  everyone.  The  committee  therefore  sum- 
marizes the  purposes  of  solid  geometry  as  follows: 

(i)  To  emphasize  and  continue  the  values  of  plane  geometry,  men- 
tioned above; 


14  NATIONAL  EDUCATION  ASSOCIATION 

(2)  To  present  a  reasonable  range  of  applications  to  the  field  of 
mensuration ; 

(3)  To  cultivate  the  power  of  visualizing  solid  forms  from  flat  drawings, 
without  entering  the  technical  domain  of  descriptive  geometry. 

SECTION  C.    SPECIAL  COURSES 

(a)  Courses  for  Different  Classes  of  Students. — 

One  of  the  topics  which  this  committee  undertook  to  consider  was  that 
of  different  courses  for  various  classes  of  students  in  the  high  schools. 

After  investigation,  it  is  the  belief  of  the  committee  that  there  should 
be  no  attempt  to  outline  such  courses.  The  syllabus  as  recommended  in 
Section  E  may  be  altered  in  special  cases  by  the  omission  of  the  theorems 
printed  in  small  type  and  by  increased  emphasis  upon  theorems  which 
admit  of  direct  practical  applications. 

The  preceding  recommendation,  together  with  the  possible  omission  of 
solid  geometry,  would  reduce  the  course  to  less  than  half  the  traditional 
length.  It  seems  probable  that  no  greater  reduction  would  be  desirable 
even  for  students  in  purely  commercial  courses,  or  indeed  in  any  course  in 
which  formal  geometry  is  a  required  subject. 

(b)  Preliminary  Courses  for  Graded  Schools. — 

A  portion  of  the  report  of  this  committee  was  to  deal  with  preliminary 
courses  to  be  undertaken  in  graded  schools. 

Recommendations. — It  is  of  the  utmost  importance  that  some  work  in 
geometry  be  done  in  the  graded  schools.  For  this  there  are  at  least  two 
very  strong  reasons.  In  the  first  place,  geometric  forms  certainly  enter  into 
the  life  of  every  child  in  the  grades.  The  subject-matter  of  geometry  is 
therefore  particularly  suitable  for  instruction  in  such  schools. 

Moreover,  the  motive  for  such  teaching  is  direct.  The  ability  to  control 
geometric  forms  is  unquestionably  a  real  need  in  the  life  of  every  individual 
even  as  early  as  the  graded  school.  For  those  who  cannot  proceed  farther 
this  need  is  pressing;  the  direct  motive  involved  compares  very  favorably 
with  any  other  direct  motive  for  work  in  the  grades.  For  those  who  are 
going  on  to  the  high  school,  the  development  of  the  appreciation  of 
geometric  forms  is  almost  an  absolute  prerequisite  for  any  future  work 
in  geometry. 

Informal  work. — It  is  quite  obvious  that  no  work  of  formal,  logical 
character  should  be  undertaken  in  the  graded  schools.  The  earliest  work 
in  geometry  will  doubtless  be  so  informal  that  it  will  not  constitute  a 
separate  course.  Instruction  in  drawing,  in  pattern-making,  and  in 
elementary  manual  training  furnishes  a  basis  for  considerable  geometric 
work  even  in  the  first  grades  of  the  primary  school. 

Such  work  as  this  should  be  encouraged,  tho  no  special  outline  of  it  can 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS  15 

be  given  on  account  of  its  dependence  upon  other  courses.  The  construc- 
tions for  erecting  perpendiculars,  bisectors  of  angles,  etc.,  can  and  should 
be  given  in  connection  with  such  manual-training  work  as  making  boxes, 
patterns,  etc.,  tho  no  technical  nomenclature  need  be  used.  In  such  work 
paper  folding  and  the  use  of  simple  instruments  should  be  encouraged, 
including  the  compasses,  the  ruler,  and  in  later  years  the  protractor  and 
squared  paper. 

Mensuration. — In  connection  with  arithmetic  much  geometric  work 
may  be  taken  up  which  is  consistent  with  the  child's  real  interests  and  life. 
Measurements  may  be  introduced  very  early  and  the  mensuration  of  simple 
forms  such  as  the  square,  rectangle,  and  triangle  need  not  be  long  delayed. 
After  this,  other  geometrical  forms  and  solids  may  be  introduced  under  the 
head  of  mensuration  even  earlier  than  is  now  customary.  In  properly 
conducted  schools,  the  students  will  become  familiar  at  the  same  time  with 
such  figures  as  the  circle,  cube,  sphere,  etc.,  in  manual  training  and  in  other 
elementary  courses,  such  as  nature-study,  geography,  etc. 

In  the  later  grades  practically  all  of  the  simple  geometric  forms  will  find 
their  place  in  arithmetic  under  the  head  of  mensuration,  in  drawing,  and  in 
manual  training. 

Work  in  the  higher  grades. — A  special  course  in  geometry  in  the  graded 
school  is  desirable,  if  at  all,  only  in  the  last  grade  or  the  last  two  grades. 
In  such  a  course  no  work  of  demonstrative  character  should  be  undertaken, 
tho  work  may  be  done  to  convince  the  student  of  the  truth  of  certain 
facts;  for  example,  by  paper  folding  or  cutting  a  variety  of  propositions 
may  be  made  evident,  such  as  that  the  sum  of  the  angles  of  a  triangle  is 
180°,  etc. 

The  theorem  just  named  is  typical  of  the  theorems  which  the  student 
should  know  as  facts  before  he  leaves  the  graded  school.  Many  others  of 
the  theorems  printed  in  black-face  type  in  the  syllabus  submitted  herewith 
may  be  taught  in  this  course  without  formal  proof. 

Theorems  as  facts. — Emphasis  should  be  laid  upon  the  facts  with  which 
the  student  is  already  familiar  thru  the  work  described  above.  The  course 
should  be  regarded  partially  as  a  classification  and  a  systematization  of  the 
knowledge  previously  acquired.  Thus,  simple  geometrical  forms  should  be 
brought  up  in  the  connection  in  which  they  have  arisen  in  the  student's 
past  experience.  In  taking  up  constructions,  explicit  mention  should  be 
made  of  the  previous  work  in  which  a  given  construction  occurred,  and 
further  practical  instances  of  the  use  of  such  constructions  should  be 
given. 

Drawing  to  scale. — Emphasis  should  be  laid  also  on  other  work  of  a 
concrete  nature  which  involves  direct  use  of  geometric  facts.  Thus  the 
propositions  concerning  the  similarity  of  triangles  should  be  introduced  by 
means  of  the  drawing  of  figures  to  scale.    Attention  should  be  called  to  the 


l6  NATIONAL  EDUCATION  ASSOCIATION 

cases  in  which  figures  have  been  drawn  to  scale  in  the  past.  The  usefulness 
and  the  necessity  of  the  operation  should  be  emphasized,  and  such  applica- 
tions as  the  drawing  of  house  plans,  the  copying  of  patterns  on  a  smaller 
scale,  etc.,  should  be  given.  The  use  of  cross-section  paper  for  this  purpose 
may  be  encouraged.  Finally  after  the  notions  involved  are  very  clear 
indeed,  and  after  actual  measurements  have  been  made  and  reduced  to 
scale,  the  precise  facts  regarding  similar  triangles  may  be  given.  This 
should  follow  and  not  precede  the  work  described  above.  The  applica- 
tions to  elementary  surveying  should  be  made,  if  possible,  in  actual  field 
work. 

Models  and  patterns. — The  situation  just  described  for  similar  triangles 
should  be  carried  out  as  far  as  possible  in  other  instances.  Thus  the 
important  theorems  on  the  measurement  of  angles  can  be  illustrated  in 
many  ways.  The  Pythagorean  theorem,  without  formal  proof,  can  be 
illustrated  and  made  real  to  every  student  by  reference  to  the  pattern  forms 
in  which  it  occurs,  the  calculation  of  distances,  and  other  real  applications. 
Finally  the  mensuration  formulas  can  all  be  given.  In  the  latter,  concrete 
illustrations  should  aboimd  and  verification  by  means  of  models  and 
measurements  upon  them  should  be  encouraged. 

Forms  of  solid  geometry. — Contrary  to  the  traditional  procedure,  the 
forms  of  solid  geometry  should  be  emphasized  even  more  than  those 
of  plane  geometry,  for  they  are  more  real  and  more  capable  of  concrete 
illustration. 

Not  only  should  formal  demonstration  be  avoided  but  also  long  lists  of 
definitions  which  tend  to  confuse  rather  than  enlighten.  Definitions  should 
be  stated  formally  only  after  the  concept  is  clearly  found  in  the  student's 
mind.  No  axiom  should  be  stated  as  such  at  any  point,  tho  frequent 
assumptions  should  be  made  without  an  attempt  at  proof.  In  all  such  cases 
care  should  be  taken  that  the  statements  made  seem  reasonable  to  the 
student  and  no  forward  step  should  be  taken  imtil  he  is  absolutely  con- 
vinced of  the  truth  of  the  statement. 

Justification. — That  such  work  is  of  vital  value  to  the  student  can 
scarcely  be  doubted;  that  it  is  absolutely  legitimate  will  probably  be 
admitted  by  all  interested  in  primary  education.  Its  value,  its  real  direct 
motives,  its  contact  with  life,  the  legitimacy  of  its  subject-matter  exceed 
incomparably  those  of  the  traditional  course  in  advanced  arithmetic.  At 
least,  the  course  in  arithmetic  may  be  vitalized  by  a  liberal  infusion  of  such 
geometric  work. 

If  such  a  course  is  not  given  in  the  grades — ^perhaps  even  tho  it  is — a 
course  of  similar  character  but  very  much  shorter  may  be  given  in  the  high 
school  before  formal  work  in  demonstrative  geometry  is  attempted.  In  any 
event  it  is  desirable  that  the  course  in  formal  geometry  should  not  proceed 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS  17 

in  its  traditional  groove  until  the  teacher  is  assured  that  the  ideas  mentioned 
above  are  thoroly  familiar  to  the  student. 


SECTION  D.     EXERCISES  AND   PROBLEMS 
DISTRIBUTION,   GRADING,   AND  NATURE   OF  EXERCISES 

(a)  Increasing  Number  of  Exercises. — There  has  been  a  growing  tend- 
ency in- the  last  two  decades  to  increase  abnormally  the  number  of  exercises 
to  be  considered  by  each  pupil  under  the  following  heads:  (i)  long  lists  of 
additional  theorems  (beyond  the  full  set  usually  given  in  the  texts),  (2) 
long  lists  of  problems  of  construction  having  at  best  remote  connection 
with  any  uses  of  geometry  within  reach  of  the  ordinary  high-school  pupil, 
(3)  long  lists  of  numerical  exercises  given  in  the  abstract,  that  is,  unrelated 
to  any  concrete  situation  familiar  to  the  pupil  or  arousing  his  interest. 

To  give  a  single  illustration  of  each: 

(i)  The  squares  of  two  chords  drawn  from  the  same  point  in  a  circle  have  the  same 
ratio  as  the  projections  of  the  chords  on  the  diameter  drawn  from  the  same  point. 

(2)  To  construct  a  triangle  having  given  the  perimeter,  one  angle,  and  the  altitude 
from  the  vertex  of  the  given  angle. 

(3)  Thru  a  point  P  in  the  side  AB  of  a  triangle  ABC,  a  line  is  drawn  parallel  to  BC 
so  as  to  divide  the  triangle  into  two  equivalent  parts.  Find  the  value  of  AP  in  terms 
of  AB. 

(b)  The  Distribution  of  Exercises. — It  is  recommended  that  there 
should  be  treated  in  connection  with  each  theorem  such  immediate  concrete 
questions  and  applications  as  are  available,  and  especially  early  in  the 
course  should  such  theorems  be  given  as  easily  lend  themselves  to  this  class 
of  exercises. 

For  example,  in  a  treatment  in  which  the  theorems  on  congruence  of 
triangles  are  placed  early,  there  is  the  opportunity  to  bring  in  at  once  the 
simplest  schemes  for  indirect  measurement  of  heights  and  distances.  Then 
later  as  similarity  of  triangles  is  taken  up,  there  is  the  chance  to  recur  to 
the  same  problems  and  let  the  pupil  see  how  the  principle  adds  power  and 
facility  in  making  indirect  measurements.  There  is  thus  a  progressive 
development  in  the  facility  for  solving  concrete  problems  along  with 
the  theory. 

This  principle  can  be  carried  out  in  many  different  lines.  For  example, 
in  connection  with  triangles,  circles,  and  squares,  there  are  many  applica- 
tions immediately  available  and  easily  found  in  tile  patterns,  window 
tracery,  grillwork,  steel  ceiling  patterns,  etc.  These  afford  fine  exercises 
in  construction  early  in  the  course,  and  are  equally  available  later  in  the 
computation  and  comparison  of  areas.    When  such  exercises  are  given  they 


l8  NA  TIONAL  EDUCA  TION  ASSOCIA  TION 

should  be  distributed  as  far  as  possible  in  connection  with  the  theorems 
used  in  the  construction  and  comparison  of  the  figures  involved. 

However,  only  the  simplest  uses  of  the  theorems  can  be  shown  in  the 
immediate  connection,  both  because  of  the  space  occupied  by  them  and  the 
danger  of  interrupting  the  continuity  of  the  theorems  by  too  many  exercises 
thrown  in  between  them,  and  also  because  most  of  these  applications  make 
use  of  various  different  theorems,  and  hence  must  come  after  certain  groups 
of  theorems,  thus  making  necessary  occasional  lists  of  problems  and  applica- 
tions scattered  thru  the  various  books,  as  well  as  sets  of  review  exercises  at 
the  end  of  each  book. 

The  whole  question  of  distribution  is  thus  to  be  determined  by  the 
relation  of  the  problems  and  applications  to  the  single  theorems  or  groups 
of  theorems  to  which  they  belong.  The  important  question  of  emphasis  in 
Section  E  of  this  report  is  best  brought  out  by  the  grouping  of  many  exer- 
cises around  the  basal  theorems. 

On  the  basis  of  distribution  we  have  all  extremes  in  the  various  texts, 
including:  (i)  the  purely  logical  presentation,  that  is,  the  continuous  chain 
of  theorems  with  practically  no  applications  in  concrete  setting  in  connection 
with  them  and  almost  none  at  the  end  of  the  books;  (2)  the  same  as  the 
foregoing,  except  that  the  long  sets  of  exercises  are  placed  at  the  end  of  each 
book,  where  they  loom  up  before  the  pupil  as  great  tasks  to  be  ground  thru, 
if,  indeed,  they  are  not  omitted  altogether;  (3)  the  psychological  presenta- 
tion in  which  the  more  difficult  exercises  either  are  postponed  to  a  later  part 
of  the  course  or  are  omitted  altogether,  and  the  easier  ones  are  brought 
into  more  immediate  connection  with  the  theorems  to  which  they  are 
related. 

The  time  and  space  made  available  by  the  third  method  of  presentation 
provide  an  opportunity  for  the  pupil  to  gain  some  acquaintance  with  the 
uses  of  the  theorems  as  he  proceeds  and  to  become  genuinely  interested  in 
the  development  of  the  subject.  The  committee  strongly  recommends  this 
latter  method  of  presentation.  In  expressing  its  disapproval  of  method 
(2),  it  is  not  to  be  understood  that  the  committee  objects  to  any  textbook 
because  it  offers  a  large  number  of  exercises,  placed  at  the  end  of  each  book, 
from  which  the  teacher  is  to  make  a  selection.  The  objection  to  (2)  should 
be  clear  from  reading  (3),  which  the  committee  approves. 

(c)  The  Grading  of  Exercises. — ^Too  much  cannot  be  said  in  favor  of 
a  large  number  of  simple  cases  rather  than  too  many  difficult  questions, 
especially  early  in  the  course,  but  also  even  thruout  the  secondary  course 
in  geometry. 

The  average  high-school  pupil  is  not  likely  to  become  adept  at  proving 
difficult  and  abstruse  theorems  independently  or  in  solving  complicated 
problems.  On  the  other  hand,  the  rank  and  file  are  bound  to  become 
discouraged  and  hopelessly  lost  in  the  so-called  "originals,"  unless  the 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS  19 

grading  is  carefully  done,  and  steps  of  difficulty  are  kept  down  to  a  very 
reasonable  lower  limit. 

The  ideal  treatment  would  seem  to  be:  (i)  to  make  a  proposition  appeal 
to  the  pupil  as  reasonable  by  simple  illustrations,  after  which  should  follow 
the  deductive  proof;  (2)  to  apply  the  theorem  to  more  difficult  situations, 
involving  problems  which  the  pupil  regards  as  interesting  and  worth  while. 
It  is  recognized  that  this  ideal  cannot  be  attained  with  reference  to  all  the 
theorems  of  geometry  but  it  is  believed  that  it  can  be  attained  in  very 
many  cases;  and,  wherever  this  is  possible,  great  interest  and  incentive  are 
given  to  the  pupil. 

As  a  matter  of  fact,  familiarity  with  the  elementary  truths  pertaining 
to  angles,  parallelograms,  and  circles,  when  consistently  tried  out  and 
seasoned  by  applications  to  numerous  comparatively  simple  and  interjesting 
geometric  forms  suggested  by  figures  which  abound  in  concrete  setting  on 
every  hand  within  reach  of  all,  is  usually  of  more  value  to  the  average 
pupil  (and  even  to  the  better  pupils)  than  is  the  study  of  a  larger  number 
of  abstract  theorems  or  problems  thru  which  they  are  often  forced.  Never- 
theless, for  the  benefit  of  the  brighter  pupils,  it  is  desirable  that  a  few 
comparatively  difficult  problems  be  given,  especially  at  the  ends  of  the 
various  books,  or  in  a  supplementary  list. 

(d)  The  Nature  of  Exercises. — This  topic  has  been  referred  to  under 
(a),  (b),  (c).  It  is  recognized  that  a  fair  proportion  of  the  traditional 
exercises  given  in  abstract  setting  should  find  a  place  in  a  course  in  geom- 
etry, but  the  committee  believes  that,  in  accordance  with  the  common 
practice  of  the  past  twenty  years,  this  class  of  exercises  has  been  magnified 
and  extended,  especially  with  reference  to  the  more  difficult  exercises, 
beyond  the  interest  and  appreciation  .of  the  average  pupil. 

The  committee  therefore  recommends  that  a  judicious  selection  of  a 
reasonable  number  of  abstract  originals  be  made  in  order  to  leave  time  for 
an  equally  reasonable  number  of  problems,  particularly  those  with  local 
coloring,  stated  in  concrete  setting. 

Since  ample  lists  of  abstract  originals  are  within  easy  reach  of  all 
teachers  of  geometry,  it  seems  unnecessary  to  supply  illustrations  of  such 
exercises.  But  as  the  teacher  must  generally  depend  upon  his  own  initiative 
to  supply  problems  in  concrete  setting,  it  seems  desirable  to  indicate  a  few 
sources  from  which  such  problems  may  be  obtained.  The  committee 
believes  that  a  reasonable  number  of  problems  of  this  character  creates  an 
interest  in  the  minds  of  the  pupils  that  reacts  strongly  in  augmenting  their 
understanding  and  appreciation  of  the  logical  side  of  the  subject.  But  it 
is  not  to  be  understood  that  the  committee  regards  these  problems  as 
practical  in  the  narrow  sense  of  the  word. 


20  NATIONAL  EDUCATION  ASSOCIATION 

SOURCES   OF   PROBLEMS 

(a)  Architecture,  Decoration,  and  Design. — Industrial  design  and 
architectural  ornament  are  replete  with  details  that  may  be  used  as  a  source 
of  supply  for  geometry  problems.  These  problems  are  of  three  kinds:  (i) 
the  problems  involved  in  the  construction  of  the  figures  themselves;  (2) 
the  demonstrations  necessary  to  establish  numerous  relations  which  are 
visible  to  the  mathematician  and  which  must  occasionally  be  assumed  by 
designers;   (3)  problems  in  computation. 

Among  the  industrial  products  that  involve  geometric  ornament  are 
tile  and  mosaic  floors,  parquetry,  Hnoleum,  oilcloth,  steel  ceilings,  orna- 
mental iron,  leaded  glass,  cut  glass,  and  the  like.  Figures  for  problems 
from  these  sources  may  be  made  from  the  cuts  in  trade  catalogs. 

Problems  based  on  architectural  ornament  are  largely  from  details  of 
Gothic  tracery  and  can  be  obtained  only  by  a  study  of  the  buildings  them- 
selves or  of  the  photographs  of  them  that  may  be  seen  in  architectural 
libraries.  Gothic  tracery  is  found  in  windows,  in  ornamental  iron,  in 
carved  stone  and  wood  on  the  outside  and  inside  of  buildings,  on  furniture, 
choir  screens,  rafters,  and  the  like,  that  abound  in  mediaeval  cathedrals  and 
churches  and  in  their  modern  imitations. 

These  problems  have  distinct  advantages.  In  many  cases  their  com- 
prehension and  solution  require  no  technical  knowledge  beyond  the 
elementary  mathematics  needed.  These  designs  abound,  are  largely  within 
the  reach  of  pupils,  and  their  use  in  the  classroom  brings  before  pupils  as 
nothing  else  can  the  beauty  and  widespread  application  of  geometric  forms. 
In  them  may  be  found  applications  of  many  topics  of  elementary  mathe- 
matics and  from  them  may  be  obtained  numerous  exercises  of  all  grades, 
from  the  simplest  to  the  most  complex.  By  their  use  it  is  possible,  there- 
fore, to  introduce  anywhere  in  the  work  problems  that  are  within  the  reach 
of  the  average  pupil  and  appeal  to  him  with  a  minimum  of  experiment, 
explanation,  discussion,  or  previous  special  preparation. 

(b)  Problems  of  Indirect  Measurement. — It  should  not  be  considered 
that  the  types  of  applications  under  (a)  are  relatively  of  greater  importance 
than  numerous  others.  Any  application  that  adds  interest  to  the  study  of 
rigorous  geometry  is  of  value.  Of  special  interest  are  all  simple  means  of 
effecting  indirect  measurements  of  distances,  such,  for  instance,  as  the 
numerous  applications  of  the  congruence  theorems  and  the  theorems  on 
similarity  of  triangles.  Here  the  teacher  will  find  much  assistance  in 
Principal  Stark's  Measuring  Instruments  of  Long  Ago.^  Again  in  consider- 
ing the  isosceles  triangle,  the  universal  leveling  instrument  (aside  from  the 
spirit  level)  offers  a  number  of  applications.  The  form  is  that  of  an  isosceles 
triangle  bisected  by  a  line  from  the  vertex. 

Many  simple  and  interesting  problems  in  indirect  measurement  are 
made  available  by  the  introduction  of  the  trigonometric  ratios,  sine,  cosine, 

*  School  Science  and  Mathematics,  Vol.  X,  pp.  48,  126. 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS  21 

and  tangent.  This  can  be  done  as  soon  as  the  theorems  on  similar  triangles 
are  known,  and  the  computation  by  measurement  of  a  two-place  table  of 
natural  functions  at  intervals  of  5°  affords,  of  itself,  a  good  drill  in  the  appli- 
cation of  these  theorems,  at  the  same  time  providing  material  for  solving 
concrete  problems  of  great  interest  to  young  pupils. 

It  may  not  be  possible  to  find  time  for  this  in  connection  with  the  usual 
course  in  plane  geometry.  Schools  that  devote  most  of  their  time  and  effort 
to  preparing  pupils  to  pass  entrance  examinations  for  college  would  certainly 
find  it  difficult  to  meet  any  added  requirement.  In  view,  however,  of  the 
omissions  suggested  by  this  committee  and  the  readjustment  of  emphasis 
on  basal  theorems,  time  may  be  found,  as  the  experience  of  an  increasing 
number  of  teachers  has  shown.  Where  this  can  be  done  it  constitutes  an 
important  step  in  the  closer  correlation  of  the  subjects  in  elementary 
mathematics.  In  any  case  only  the  natural  functions  should  be  used,  and 
the  applications  should  be  limited'  to  those  involving  right  triangles. 

(c)  Other  Sources. — Problems  may  be  obtained  also  from  physics, 
mechanics,  and  other  sciences,  from  engineers'  and  builders'  manuals  and 
works  on  carpentry  and  masonry,  as,  for  instance,  problems  derived  from 
various  common  forms  of  trusses  and  the  construction  of  arches.  But  there 
is  a  danger  in  connection  with  problems  from  these  sources,  that  aside  from 
the  geometry  involved,  they  may  contain  technical  terms  and  mechanical 
terms  and  mechanical  features  unknown  to  the  average  pu^il  and  not  easily 
understood  without  more  explanation  and  consequent  distraction  from  the 
geometry  itself  than  is  warranted  in  the  ordinary  course.  Problems  of  this 
class  should  be  carefully  tried  out  and  sifted  before  being  adopted  for  use. 

(d)  Illustrative  Problems. — There  are  given  below  a  few  typical  prob- 
lems which  are  suggested  for  the  purpose  of  making  clear  what  the  com- 
mittee has  in  mind.  Thru  simple  problems  of  these  types  and  many  others 
which  might  be  suggested  much  interest  can  be  imparted  to  the  study  of 
demonstrative  geometry,  even  tho  the  problems  be  not  practical  in  the 
strict  sense  of  the  word.  The  danger  of  using  too  many  problems  in  any 
narrow  field  is,  however,  apparent. 

(i)  The  theorem  regarding  the  angle  sum  in  a  triangle  has  a  large  number  of  applica- 
tions.    For  example,  to  measure  PC,  stand  at  some  convenient 

point  A  and  sight  along  APC  and  (by  the  help  of  an  equilateral 

triangle  cut  from  pasteboard)  along  AB.    Then  walk  along  AB 

until  a  point  B  is  reached  from  which  BC  makes  with  BA  an 

angle  of  the  equilateral  triangle  (60**).    Then  AC=AB,  and  since 

AP  can  be  measured  we  can  find  PC.    This  is  an  example  of  a 

problem  that  adds  interest  to  the  work  without  being  itself  a  practical  application  that 

would  be  used  by  a  surveyor. 

(2)  A  problem  of  the  same  nature  is  the  following:  To  measure  AC,  first  measure 
the  angle  CAX,  either  in  degrees  with  a  protractor  or  by  sight- 
ing across  a  piece  of  paper  and  marking  it  down.  Then  walk 
along  XA  produced  until  a  point  B  is  reached,  from  which  BC 
makes  with  BA  an  angle  equal  to  half  of  angle  CAX.  Then  it 
is  easily  shown  that  AB=AC. 


22 


NATIONAL  EDUCATION  ASSOCIATION 


B 


(3)  The  sailor  makes  use  of  this  principle  when  he  "doubles  the  angle  on  the  bow" 
to  find  his  distance  from  a  hghthouse  or  promontory. 
If  he  is  saiUng  on  the  course  ABC  and  he  notes  a  light- 
house L  when  he  is  at  A,  and  takes  the  angle  A,  and  if 
he  notices  when  the  angle  that  the  lighthouse  makes 
with  his  course  is  just  twice  the  angle  noted  at  A, 
then  BL=AB.  He  has  AB  from  his  log,  so  he  knows 
the  distance  BL. 

(4)  To  measure  the  line  XY,  when  the  observer  is  at  A,  we  may  measure  any  line  AB 
along  the  stream.  Then  the  observer  may  take  a  carpenter's  square,  or  even  a  large 
book,  and  walk  along  AB  until  a  point  P  is  reached  from 
which  X  and  B  can  be  seen  along  two  sides  of  the  square. 
Similarly  the  point  Q  may  be  fixed.  Then  by  walking 
along  YM  to  a  point  Y'  that  is  exactly  in  hne  with  M 
and  Y  and  also  with  P  and  X,  the  point  Y'  is  fixed. 
Similarly  X'  is  fixed.    Then  X' Y' = XY. 

(5)  A  field  containing  9  acres  is  represented  by  a 
triangular  plan  whose  sides  are  12  in.,  17  in.,  and  25  in. 
drawn  ? — Conant. 

(6)  Assuming  the  earth  to  be  a  sphere  of  which  the  radius  is  3,960  miles,  find  the 
length  of  one  degree  of  longitude  at  60"  north  latitude,  and  compare  its  length  with  that 
of  one  degree  of  longitude  at  the  equator. 

(7)  ABCD  is  a  square.    Equal  distances  AE,  BF,  CG,  and  DH  are  measured  oflE  on 


On  what  scale  is  the  plan 


the  sides  AB,  BC,  CD,  and  DA  respectively.    If  the  lines  AF,  BG,  CH,  and  DE  are 
drawn  intersecting  at  Y,  Z,  W,  and  X,  prove  that  XYZW  is  a  square.    If  AB =a,  and  AE 


is  \  of  AB,  prove  that  AF=-i/io, 
the  area  of  XYZW  is  — • 


AX=— r/I5,  XY= 
10 


V^io,  FY= — i/io;  and  that 


30 


The  solution 


This  figure  is  the  basis  of  an  Arabic  design  used  for  parquet  floors, 
involves  both  algebraic  and  geometric  work  in  concrete  setting. 

(8)  ABC  is  an  equilateral  arch,  and  CD  its  altitude.  A  is  the  center  of  the  arc  BC 
and  B  the  center  of  the  arc  AC.  The  equilateral  arches  AED  and  DFB  are  erected  on 
AD  and  BD  respectively.  D  is  the  center  of  arc  AE  and  FB,  and  A  and  B  are  centers  of 
arcs  ED  and  DF,  each  drawn  with  ^AB  as  radius.  What  is  the  locus  of  centers  of  circles 
tangent  to  CA  and  CB  i^^To  ED  and  DF  Pr.'^To  AC  and  DF  Pc^To  CB  and  ED  Po^Con- 
struct  a  circle  tangent  to  the  arcs  AC,  CB,  ED,  and  FD.  e  o 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS 


23 


This  figure  is  the  basis  of  a  common  Gothic  window  design.    The  solution  involves 
the  intersection  of  loci. 


,  (9)  CD  is  the  perpendicular  bisector  of  AB.  Equal  distances  AX  and  BY  are 
measured  off  on  AD  and  BD  respectively.  EF  is 
perpendicular  to  CD  at  C.  Circles  are  drawn  with  X 
and  Y  as  centers  and  AX  and  BY  as  radii.  Construct 
a  circle  tangent  to  EF  at  C  and  to  circle  X.  Prove 
that  this  circle  is  also  tangent  to  the  circle  Y. 

If  CD  is  less  than  |AB  the  part  of  this  figure 
between  lines  CD  and  AB  is  one  form  of  a  three 
centered  arch. 


\ 


(10)  In  the  drawing  below,  which  is  the  basis  of  the  mosaic  floor  design  to  the  right, 
the  circle  with  center  N  is  inscribed  in  one  of  the  squares  whose  side  is  SH.    The  arcs 


\ 

^f 

^^^}J 

■"7 

^iV\ 

F^^J^ 


ORQ  and  OKL  are  drawn  with  the  vertices  of  the  square  as  centers  and  half  the  side  as 
radius.  The  semicircles  LMN  and  NPQ  are  drawn  on  LN  and  NQ  as  diameters.  Find 
the  areas  of  the  various  figures  bounded  by  circular  arcs  within  this  square.  Note  the 
symmetry  of  the  whole  figure  within  the  square  AB  CD . 

(11)  ABC  is  an  equilateral  triangle.    A  and  B  are  centers  of  arcs  BC  and  AC  respec- 
tively.   CD  is  the  altitude  of  triangle  ABC.    Arcs  DF  and  DE,  constructed  with  radii 


24 


NATIONAL  EDUCATION  ASSOCIATION 


equal  to  AB,  are  tangent  to  CD  at  D  and  intersect  AC  and  CB  respectively  at  E  and  F. 
Construct  a  circle  tangent  to  the  arcs  DE,  DF,  AC,  and  BC. 


Suppose  the  problem  solved.  Let  O  be  the  center  of  the  circle.  Connect  O  with  B, 
the  center  of  arc  AC,  and  with  H,  the  center  of  arc  DE.  From  triangles  ODB  and  OHB 
the  following  equation  is  derived: 

{s-ry-{s/2y  =  {s^-ry-s', 
where  s  is  the  length  of  AB  and  r  is  radius  of  the  required  circle. 

This  figure  is  the  basis  of  a  church  window  design.  Many  problems  of  this  type  may 
be  easily  obtained. 

(12)  A  quarter-mile  running-track  has  two  parallel  sides  and  semicircular  ends. 
Each  straightaway  section  is  equal  in  length  to  one  of  the  ends.  If  the  track  measures 
exactly  one-fourth  of  a  niile  at  the  curb,  or  inner  edge,  how  much  distance  does  a  runner 
lose  in  running  two  feet  from  the  curb  ?  Six  feet  ?  What  is  the  area  of  the  track  if  it  is 
15  feet  wide?  What  is  the  area  of  the  inclosed  field?  What  are  the  dimensions  of  a 
rectangular  field  sufl5ciently  large  to  contain  such  a  track  ?  What  will  it  cost  at  $2 .  00 
per  cubic  yard  to  cover  such  a  track  with  cinders  to  a  depth  of  2  inches  ? — Pettee. 

(e)  References  to  Sources  of  Problems. — In  connection  with  the  recent 
search  for  real  applied  problems  in  elementary  mathematics  numerous 
bibliographies  have  been  compiled  to  which  reference  is  here  made,  as  well 
as  to  a  few  other  books,  aside  from  current  texts,  which  may  be  helpful  to 
teachers.  Concrete  problems  should  be  selected  carefully  and  used  wisely. 
Those  which  may  appear  to  one  class  of  pupils  as  real  applied  problems  may 
seem  highly  abstract  to  another  class.  Probably  few  problems  in  the 
following  lists  would  appear  real  to  all  pupils  and  yet  all  are  likely  to  find 
increased  interest  in  any  problem  which  has  a  concrete  origin. 
Printed  bibliographies. — 

(i)  A  list  of  ^S  titles  of  books  and  21  titles  of  trade  journals.  School 
Science  and  Mathematics ^  Vol.  IX,  No.  8,  1909,  pp.  788-98. 

(2)  A  more  extended  list  of  books  on  the  whole  range  of  applied  prob- 
lems, School  Science  and  Mathematics,  Vol.  VIII,  No.  8,  November,  1908, 
pp.  641-44. 

From  this  list  Saxelby,  Godfrey  and  Siddons,  and  Perry  may  be  men- 
tioned especially. 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS  25 

(3)  A  comprehensive  list  of  books  and  journals  relating  to  the  uses  of 
geometry  in  architecture,  decoration,  and  design  in  a  new  volume  entitled 
A  Source  Book  of  Problems  for  Geometry,  by  Mabel  Sykes.  Allyn  and 
Bacon,  Boston,  191 2. 

(4)  A  vast  bibliography  of  suggestive  titles,  with  a  classification  and 
discussion  of  some  phases  of  industrial  problems,  by  a  Committee  of  the 
National  Education  Association  on  the  Place  of  Industries  in  Public 
Education,  Proceedings  of  the  Association,  1910,  pp.  652-788. 

Collections  of  problems. — 

(i)  ''Real  Problems  in  Geometry,"  Teachers  College  Record,  March,. 
1909.  A  classification  and  discussion  of  types  of  applied  problems,  by 
James  F.  Millis. 

(2)  "Real  Applied  Problems  in  Algebra  and  Geometry,"  School  Science 
and  Mathematics.  A  collection  begun  in  1909  by  a  committee  of  the  Central 
Association  of  Science  and  Mathematics  Teachers.  The  work  is  still  in 
progress.  The  problems  collected  up  to  November,  1909,  have  been 
classified  and  published  in  pamphlet  form. 

Other  selected  titles. — 

(i)  Lessons  in  Experimental  Geometry,  tHall  and  Stevens.  The 
Macmillan  Company,  New  York,  1905. 

(2)  Numerical  Problems  in  Geometry,  J.  G.  Estill.  Longmans,  Green 
and  Company,  New  York,  1908. 

(3)  Mensuration,  G.  B.  Halsted.     Ginn  and  Company,  Boston,  19 10. 

(4)  Elementary  Mensuration,  F.  H.  Stevens.  The  Macmillan  Company, 
New  York,  1908. 

(5)  A  Notebook  of  Experimental  Mathematics,  Godfrey  and  Bell. 
Edward  Arnold,  London,  1905. 

(6)  Elements  of  Mechanics,  M.  Merriman.  John  Wiley  and  Sons, 
New  York,  1905. 

(7)  Shop  Problems  in  Mathematics,  Breckenridge,  Mersereau,  and  Moore. 
Ginn  and  Company,  New  York,  19 10. 

(8)  Pocket  Companion  containing  Tables,  etc.  Carnegie  Steel  Company,. 
Pittsburgh,  Pa.,  1903. 

(9)  Leitfaden  der  Geometric,  Jahne  and  Barbisch.    Vienna,  1907. 

(10)  Raumlehre  fur  Mittelschulen,  Martin  and  Schmidt.     Berlin,  1898.. 

(11)  Geometrie  fur  die  Zwecke  des  practischen  Lebens,  G.  Ehrig.  Leipzig, . 
1906. 

(12)  Mathematische  Aufgaben,  Schulze  and  Pahl.     Leipzig,  1908. 

(13)  Cours  abregee  de  Geometrie,  Bourlet  and  Baudoin.     Paris,  1907. 

(14)  Cahiers  d'execution  de  dessins  geometriques,  M.  P.  Baudoin.     Paris.. 

(15)  Geometria  Intuitiva,F.'P2isqua\i.     Milan. 

(16)  Regole  di  Geometria  Pratica,  F.  Andreotti.    Florence,  1897. 

(17)  The  Power  of  Form  Applied  to  Geometrical  Tracery,  R.  W.  Billings. 
London,  185 1. 


26  NATIONAL  EDUCATION  ASSOCIATION 

(i8)  Gothic  Architecture  in  England,  Francis  Bond.  B.  T.  Botsford, 
London,  1905. 

(19)  Les  elements  de  Vart  arabCy  Jules  Bourgoin.     Paris,  1879. 

(20)  Pattern  Design,  Lewis  F.  Day.  B.  T.  Botsford,  London;  Scribner's 
Sons,  New  York,  1903. 

(21)  Geometrische  Ornamentikj  L.  Diefenbach.     Max  Spielmeyer,  Berlin. 

(22)  Romano-British  Mosaic  Pavements,  Thomas  Morgan.  London, 
1886. 

(23)  Decorated  Windows,  A  Series  of  Illustrations,  Edmund  Sharpe. 
London,  1849. 

(24)  Specimens  of  Tile  Pavements,  Henry  Shaw.     London,  1858. 

(25)  Specimens  of  Geometrical  Mosaics  of  the  Middle  Ages,  Sir  Matthew 
Wyatt.     London,  1848. 

(26)  The  Teaching  of  Geometry,  David  Eugene  Smith.  Ginn  and  Com- 
pany, Boston,  191 1. 

PROBLEMS  INVOLVING  LOCI 

(a)  Phraseology. — While  the  committee  does  not  wish  to  prescribe  the 
exact  phraseology  of  any  definition,  it  would  recommend  greater  care  in  the 
formulation  of  the  definitions  underlying  the  subject  of  loci.  It  is  suggested 
that  any  definition  used  should  be  substantially  equivalent  to  the  following: 

The  locus  of  a  point  (or  the  locus  of  points)  satisfying  given  conditions  is  a  configura- 
tion such  that: 

(i)  All  points  lying  on  the  configuration  satisfy  the  conditions; 
(2)  All  points  satisfying  the  conditions  lie  on  the  configuration. 

It  would  seem  desirable  to  make  all  proofs  on  loci  conform  to  this 
definition.  It  is  of  course  understood  that  the  teacher  will  lead  the  pupil 
up  to  such  a  definition  thru  varied  forms  of  concrete  description,  such  as 
"path  of  a  point  in  motion,"  etc. 

(b)  Motion  in  Geometry. — ^It  seems  well  to  give  some  consideration  to 
the  place  of  motion  in  a  well-rounded  course  in  elementary  geometry,  and 
to  bear  in  mind  that  this  course  is  all  the  geometry  to  be  studied  by  the 
majority  of  high-school  pupils.  It  has  recently  been  urged  by  prominent 
European  mathematicians  that  motion  should  be  given  a  more  prominent 
place  at  this  stage.  We  may  well  recall  that  the  space  concepts  dealt  with 
in  our  usual  courses  in  geometry  are  almost  entirely  to  be  described  as 
static.  There  is  in  theorems  and  problems  on  loci  a  dynamic  element 
that  is  of  importance.  The  pupil  is  pretty  familiar  with  motion  as  a  con- 
crete experience,  and  it  seems  of  first-class  importance  to  idealize  some 
such  concrete  experiences,  until  they  possess  the  precision  of  geometry. 

For  example,  in  a  given  plane  we  may  consider  in  a  way  well  described 
as  a  static  configuration  the  perpendicular  bisector  of  a  line-segment  joining 
two  points;  but  when  we  consider  this  line  as  generated  by  a  point  moving 
in  the  plane  in  such  a  way  that  it  is  always  equidistant  from  the  two  given 
points,  we  add  a  dynamic  element. 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS  27 

As  to  phraseology,  the  expression  "locus  of  points"  suggests  a  static 
configuration,  while  the  expression  "locus  of  a  point"  emphasizes  the 
dynamic  element  and  is  equivalent  in  thought  to  the  "path  of  a  point  mov- 
ing with  certain  prescribed  conditions."  Both  of  these  phases  should  have 
a  place  in  the  treatment  of  loci  problems,  and  thus  both  forms  of  expression 
should  be  used,  the  one  or  the  other  being  more  suggestive  in  different  cases. 
It  is  even  desirable  to  use  different  forms  in  describing  a  given  case  to  make 
clear  the  idea  and  to  cultivate  facility  in  expression. 

(c)  Concrete  Nature  of  Loci. — Contrary  to  the  usual  conception,  the 
locus  idea  is  one  that  may  very  easily  be  made  concrete  and  brought  down 
to  the  comprehension  of  young  pupils.  For  example,  the  opening  of  a 
book  or  of  a  door  suggests  a  variety  of  loci.  The  same  may  be  said  of 
many  concrete  illustrations  easily  accessible  to  the  pupil. 

In  this  way,  loci  problems  may  and  should  be  introduced  at  certain 
stages  of  the  subject.  For  example,  in  Book  I:  The  locus  of  a  point  equi- 
distant from  two  fixed  points,  equidistant  from  two  intersecting  lines,  or 
from  two  parallel  lines,  or  at  a  given  distance  from  a  fixed  line.  In  the  book 
on  circles,  the  locus  of  all  points  equidistant  from  a  fixed  point,  the  locus  of 
the  centers  of  circles  of  fixed  radius  and  tangent  to  a  given  line,  the  locus 
of  the  centers  of  all  circles  tangent  to  two  parallel  lines  or  two  intersecting 
lines,  and  the  locus  of  the  vertices  of  all  triangles  having  a  common  base 
and  equal  vertex  angles.  In  solid  geometry,  the  locus  of  points  equidistant 
from  a  given  point,  from  two  given  points,  from  a  given  plane,  from  two 
intersecting  planes,  from  two  parallel  planes,  etc. 

(d)  Loci  in  Problems  of  Construction. — Important  features  of  the 
construction  problems  in  geometry  are  dependent  upon  loci  considerations 
which  should  be  emphasized  in  this  connection.     For  example: 

(i)  To  find  the  locus  in  the  plane  of  all  points  equidistant  from  three 
given  points,  it  is  necessary  to  determine  the  intersection  of  two  loci  both 
of  which  are  straight  lines. 

(2)  To  find  the  locus  in  the  plane  of  all  points  equidistant  from  a  fixed 
point  and  at  a  given  distance  from  a  given  line,  it  is  necessary  to  find  the 
intersection  of  two  loci  one  of  which  is  a  straight  line  and  the  other  a  circle* 

The  discussion  of  the  various  possibilities  in  connection  with  such 
problems  is  one  of  the  most  valuable  exercises  for  the  pupil.  For  example,, 
as  to  whether  there  are  one,  two,  or  no  points  fulfilling  the  conditions  in  the 
second  example  above.  While  it  may  be  possible  to  solve  and  discuss  such 
problems  without  using  the  term  locus  at  all,  yet  this  leads  to  round-about> 
and  awkward  explanations,  while  the  language  of  loci  is  elegant  and  concise. 

Moreover,  facility  in  the  use  of  this  language  is  not  only  desirable  from 
the  standpoint  of  the  high-school  pupil  but  is  of  the  utmost  importance  for 
those  who  may  continue  the  study  of  geometry  in  college. 

(e)  To  Summarize. — ^The  locus  idea  is  deserving  of  a  careful  and  sys- 
tematic treatment  for  the  following  reasons: 


28  NATIONAL  EDUCATION  ASSOCIATION 

(i)  It  introduces  a  dynamic  element  thru  the  consideration  of  the  idea 
of  motion. 

(2)  It  presents  an  elegant  language  for  the  statement  of  those  propo- 
sitions on  which  nearly  all  of  our  problems  of  construction  are  based. 

(3)  It  aids  greatly  in  the  cultivation  of  space  intuition  and  in  emphasiz- 
ing the  important  concept  of  functionality. 

(f)  Additional  Illustrations  Appropriate  for  Use. — 

1.  Find  the  locus  of  all  points  at  a  fixed  distance  from  the  sides  of  a  triangle,  always 
measuring  from  the  nearest  point  of  a  side. 

2.  Find  the  locus  of  points  such  that  the  sum  of  the  squares  of  the  distances  from 
two  lines  intersecting  at  right  angles  is  100. 

3.  Find  the  locus  of  the  vertices  of  a  regular  polygon  of  a  given  number  of  sides  that 
can  be  circumscribed  about  a  given  circle. 

4.  Find  the  locus  of  the  midpoints  of  the  sides  of  regular  polygons  of  a  given  number 
of  sides  that  can  be  inscribed  in  a  given  circle. 

5.  Find  the  locus  of  all  points  from  which  a  given  line-segment  subtends  a  given  angle. 

6.  Find  the  locus  of  a  point  the  sum  of  the  squares  of  whose  distances  from  two  given 
points  is  constant. 

7.  Find  the  locus  of  a  point  the  difference  of  the  squares  of  whose  distances  from  two 
given  points  is  constant. 

8.  Find  the  locus  of  all  lines  drawn  thru  a  given  point,  parallel  to  a  given  plane. 

9.  Find  the  locus  of  a  point  in  space  equidistant  from  three  given  points  not  in  a 
straight  line. 

ALGEBRAIC   METHODS   IN   GEOMETRY 

The  committee  feels  that  the  use  of  algebraic  forms  of  expression  and 
solution  in  the  geometry  courses  may  well  be  extended,  with  advantage  to 
both  algebra  and  geometry,  and  that  this  may  be  done  without  in  any  way 
encroaching  upon  the  field  of  analytic  geometry,  which  belongs  to  a  later 
stage  of  development. 

(a)  The  Notation  Should  Be  More  Algebraic. — While  it  is  not  feasible 
or  desirable  to  lay  down  hard-and-fast  rules  to  standardize  the  notation  of 
geometry,  an  examination  of  current  texts  makes  it  evident  that  some 
notations  in  common  use  are  unnecessarily  awkward  when  compared  with 
the  notations  used  in  elementary  algebra.  The  notation  of  geometry  is,  in 
general,  improved  by  much  use  of  lower-case  letters  to  represent  numerical 
values,  leaving  capitals  to  represent  points.  This  notation  is  here  called 
algebraic  because  the  student  will  recognize  the  relations  of  equality  and 
inequality  much  more  readily  in  the  familiar  notation  of  algebra  than  if 
these  relations  are  presented  in  a  notation  not  used  in  algebra. 

(b)  Algebraic  Statement  of  Propositions. — Many  of  the  theorems 
of  geometry  may  be  stated  to  advantage  in  algebraic  form,  thus  giving 
definiteness  and  perspicuity  and  especially  emphasizing  the  notion  of 
functionality.  This  mode  of  expression  can  be  made  of  much  value  to  the 
student  if  he  is  required  to  translate  into  English  all  the  symbols  involved. 

^        The  following  are  illustrations  of  the  algebraic  statement  of  propositions : 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS  29 

(i)  In  any  triangle,  a  =  hh/2,  where  a  is  the  area,  h  is  the  base  and  h  is  the  altitude. 

(2)  In  a  right  triangle,  c^  =  a*+6^  where  c  is  the  hypotenuse,  and  a  and  b  are  the 
sides  including  the  right  angle. 

(3)  In  any  triangle  c^  =  a^-{-b'=i=2ap,  where  a,  h,  c  are  sides  of  the  triangle  and  p  is 
the  projection  of  h  on  a. 

(4)  For  any  secant  and  tangent  drawn  from  a  point  to  a  circle,  we  have  f^^sx,  where 
t  is  the  length  of  the  tangent,  s  is  the  length  of  the  secant,  and  x  is  the  length  of  the 
external  part. 

It  is  not  intended  to  convey  the  impression  that  the  usual  statement  of 
propositions  should  be  replaced  by  the  algebraic  statements  but  rather  that 
the  student  should  be  required  to  translate  the  one  form  of  statement  into 
the  other.  The  algebraic  statements  are  often  superior  to  the  usual  state- 
ments in  point  of  brevity  and  conciseness.  Moreover,  the  algebraic 
statement  prepares  for  the  idea  of  functionality,  which  is  too  little  under- 
stood by  persons  who  are  not  trained  in  mathematics  beyond  the  high-school 
course.  That  is  to  say,  some  appreciation  of  the  influence  of  changing  one 
part  of  a  configuration  on  other  parts  of  the  configuration  can  often  be 
gained  readily  from  the  algebraic  statement. 

(c)  Geometrical  Construction  of  Formulas. — Some  propositions  can 
be  proved  simply  and  elegantly  by  methods  involving  algebra.  It  is 
somewhat  usual  in  textbooks  on  geometry  to  give  a  proof  of  the  geometrical 
statement  of  such  an  algebraic  formula  as  {a-Yhy  =  a^-\-h^-\-2ab,  where  a 
and  b  are  the  numerical  measures  of  the  line-segments,  but  to  neglect  the 
geometrical  construction  of  the  formula.  The  latter  seems  to  be  the  point 
of  greatest  importance.  It  is  not  additional  evidence  of  the  validity  of  the 
theorem  that  is  sought.  That  is  established  in  algebra.  What  is  of  first- 
rate  importance  is  to  give  a  geometrical  picture  of  the  formula,  thus  showing 
a  certain  geometrical  interpretation  and  to  have  the  student  put  the  result 
into  geometrical  phraseology  when  a  and  h  are  line-segments. 

The  construction  of  line-segments  a-\-b,  a—b,  and  of  areas  ab,  (a-\-by, 
(a— by,  where  a  and  b  are  line-segments,  should  come  early  in  the  course. 
Later,  when  the  requisite  theorems  are  being  developed,  the  further  elemen- 
tary expressions 

ka,  \,  — ,  Vaby~aH^^  Va'-b',  aVk, 
k    c 

where  a,  b,  and  c  are  line-segments  and  ^  is  a  positive  integer,  should  be 
constructed. 

This  interdependence  of  algebra  and  geometry  is  a  matter  of  no  small 
importance  both  historically  and  for  subsequent  mathematical  work.  It 
should  be  brought  out  by  suitable  exercises  that  the  use  of  algebra  often 
enables  one  to  establish  relations  from  which  a  geometrical  construction 
can  be  made  readily  or  to  show  the  nature  of  a  difficulty  involved. 

For  example,  to  inscribe  a  square  in  a  semicircle: 

If  X  represents  the  side  of  the  square  and  r  the  radius  of  the  circle, 


30  NATIONAL  EDUCATION  ASSOCIATION 

we  have  at  once  from  a  right  triangle  that  r^  =  x^+^V4  2,nd  hence 
x=  =^  ii/5r,  which  can  be  constructed  from  exercises  given  above. 

(d)  Geometric  Exercises  for  Algebraic  Solution. — Some  exercises  for 
algebraic  solution,  such  as  are  found  in  many  recent  texts,  should  find  a 
place  in  any  course  in  geometry.  For  example,  the  following  is  a  suitable 
exercise  after  the  proposition  stating  that  a  =  bh,  where  a  is  the  area,  b  and 
h  are  sides  of  a  rectangle: 

The  area  of  a  rectangle  is  480  square  inches.  Each  side  of  the  rectangle  is  increased 
I  inch,  and  by  this  change,  the  area  is  increased  45  square  inches.  Find  the  sides  of 
the  rectangle. 

Similarly,  after  the  proposition  pertaining  to  secants  and  tangents  to  a 
circle,  the  following  is  suitable: 

A  secant  line  which  passes  thru  the  center  of  a  circle  of  radius  10  is  intersected  by  a 
tangent  of  length  15.     Find  the  length  of  the  external  part  of  the  secant. 

Such  exercises  do  much  to  unify  geometry  and  algebra,  and  may  well 
replace  some  of  the  usual  exercises. 

Finally,  after  the  theorem  on  the  volume  of  a  frustum  of  a  pyramid,  a 
problem  like  the  following  has  value  as  an  algebraic  exercise,  altho  it  is  in  no 
sense  a  real  applied  problem. 

A  pier  is  built  of  solid  concrete  construction,  in  the  form  of  a  frustum  of  a  pyramid 
with  square  bases.  The  altitude  is  twice  an  edge  of  the  lower  base  and  the  area  of  the 
lower  base  is  four  times  that  of  the  upper  base.  Find  the  dimensions  of  each  base  if  the 
pier  contains  600  cubic  feet  of  solid  concrete. 

SECTION  E.    SYLLABUS  OF  GEOMETRY 
PREFACE   TO   LISTS   OF   THEOREMS 

(i)  Lists  not  exhaustive. ^The  lists  of  theorems  which  follow  are  not  to 
be  taken  as  exhaustive,  and  it  is  distinctly  understood  that  theorems  may 
be  added  at  the  discretion  of  the  teacher.  For  example,  the  theorem  on 
the  existence  of  regular  polyhedra  may  find  a  place  in  certain  courses. 
Some  theorems  are  omitted  only  with  the  understanding  that  they  may  be 
inserted  as  exercises  for  the  student;  some  such  possible  exercises  are: 

In  any  triangle,  the  product  of  any  two  sides  is  equal  to  the  product  of  the  segments 
of  the  third  side  formed  by  the  bisector  of  the  opposite  angle,  plus  the  square  of  the 
bisector. 

The  medians  of  a  triangle  meet  in  one  point  which  divides  each  median  in  the  ratio  i :  2. 

To  divide  a  given  straight  line-segment  in  extreme  and  mean  ratio. 

To  find  the  area  of  a  triangle  in  terms  of  its  sides. 

To  construct  a  square  having  a  given  ratio  to  a  given  square. 

The  surface  of  a  sphere  is  equivalent  to  the  area  of  four  great  circles. 

(2)  Logical  order. — Altho  there  is  some  indication  of  a  possible  order  in 
the  lists,  there  is  no  intention  of  specifying  any  definite  order.     It  would  be 
impossible  to  carry  out  as  a  whole  precisely  the  order  stated  below. 
^  In  several  connections  the  words  "corollary  to"  or  "synonymous  to" 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS  31 

may  seem  to  imply  an  order.  These  phrases  are  used  only  to  indicate  the 
reason  for  putting  the  theorem  quoted  in  the  group  in  which  it  appears. 
Thus  in  Group  II  on  p.  41,  it  would  not  be  clear  in  every  case  that  each 
theorem  is  a  corollary  of  plane  geometry  without  such  a  suggestion  of 
possible  derivation. 

It  should  be  noticed  that  some  logical  arrangements  would  necessitate 
the  insertion  of  the  theorems  omitted  in  this  list.  Such  an  insertion  is 
entirely  in  the  spirit  of  this  report,  as  is  also  any  conceivable  change  in  the 
order,  except  where  specified  explicitly  in  the  report. 

(3)  Subsidiary  theorems. — ^A  number  of  theorems  omitted  in  the  lists 
below  may  well  be  given  as  ordinary  statements  in  the  course  of  the  text  as 
corollaries,  or  as  remarks,  without  the  emphasis  which  attaches  to  formal 
theorems.  Among  such  general  statements  which  should  by  all  means  be 
made  at  the  proper  points  are  the  following  : 

No  triangle  can  have  more  than  one  right  angle  or  more  than  one  obtuse  angle. 

The  third  angle  of  a  triangle  can  be  found  if  two  are  known. 

An  equilateral  triangle  is  equiangular. 

The  square  on  a  side  of  a  right  triangle  adjacent  to  the  right  angle  is  equal  to  the 
square  on  the  hypotenuse  minus  the  square  on  the  other  side. 

Thru  three  points  not  in  a  straight  line  not  more  than  one  plane  can  be  passed. 

The  areas  of  two  spheres  are  to  each  other  as  the  squares  of  their  radii;  their  volumes 
as  the  cubes  of  their  radii.     (Like  statements  for  other  solids.) 

The  number  of  such  statements  is  exceedingly  large  and  all  of  them 
•  could  not  be  given  in  any  syllabus.  A  large  majority  are  at  the  present  time 
stated,  if  at  all,  in  the  course  of  the  reading-matter,  or  in  exercises,  and  not 
as  explicit  theorems.  It  is  understood,  and  indeed  expected,  that  these 
statements,  together  with  many  which  are  omitted  from  the  lists  of  theorems 
below,  should  be  treated  in  this  manner. 

(4)  Informal  proofs. — ^The  theorems  given  below  under  the  heading 
"Theorems  for  Informal  Proofs"  should  be  stated  at  the  proper  points  in 
the  text  and  in  theorem  form,  or  as  postulates.  Their  proofs,  however,  can 
well  be  omitted  where  this  omission  is  suggested,  or  be  made  exceedingly 
informal  by  the  insertion  of  a  single  phrase  which  will  give  the  proper 
suggestion  for  the  proof.  Many  other  theorems  which  are  equally  obvious 
are  not  stated  because  they  occur  more  naturally  as  corollaries  or  as  exer- 
cises.    (See  the  preceding  paragraph.) 

Regarding  the  method  of  proof  in  general,  while  the  demonstrations 
should  remain  as  logical  as  they  are  at  present,  it  is  suggested  that  the 
formalities  of  logic,  as  such,  be  frequently  dispensed  with  to  a  very  con- 
siderable extent  and  that  the  propositions  be  frequently  stated  and  proved 
in  language  resembling  that  to  be  found  in  any  other  mathematical  text- 
book. This  is,  indeed,  the  style  of  many  classical  treatises,  such  as 
Legendre's  or  Euclid's.  It  is  certainly  satisfactory  and  there  is  no  reason 
why  the  proof  should  not  remain  quite  as  logical  when  the  older  style 
is  followed. 


32  NATIONAL  EDUCATION  ASSOCIATION 

The  symbolic  form  of  demonstration  which  appears  in  many  texts 
should  be  regarded  simply  as  a  shorthand  expression  of  a  complete  proof  in 
ordinary  English  phraseology.  The  latter  should  be  given  by  the  student 
in  all  cases.  The  ability  to  pass  from  the  symbolic  form  to  ordinary  English, 
that  is,  to  translate  the  shorthand  into  the  language  of  everyday  life,  should 
be  constantly  tested  by  the  teacher,  for  the  same  reason  that  the  formulas 
of  algebra  derive  their  real  meaning  and  power  from  the  thought  content 
which  the  student  can  attach  to  them. 

(s)  Arrangement  for  emphasis. — The  main  list  of  theorems  is  divided 
into  several  heads,  each  group  being  introduced  by  a  theorem  of  suitable 
importance  upon  which  the  rest  of  the  theorems  in  that  group  depend  more 
or  less  closely.  This  arrangement  has  been  selected  in  order  to  emphasize 
the  importance  of  a  few  major  propositions,  namely,  those  which  carry  a 
maximum  of  applications  and  from  which  the  rest  can  be  derived,  thus 
serving  as  a  nucleus  for  the  whole  of  geometry. 

This  effort  to  gain  emphasis  has  been  carried  out  still  farther  by  printing 
the  theorems  in  different  grades  of  type  so  that  those  of  fundamental 
importance  and  of  basal  character  are  printed  in  black-face  type;  those  of 
considerable  importance  which  are  secondary  only  to  the  preceding  ones  are 
printed  in  italics.  A  number  of  other  theorems  are  printed  in  roman  type, 
while  the  least  important  are  printed  in  small  type.  The  latter  (small- type 
theorems)  may  be  omitted  without  serious  danger,  or  they  may  be  used  as 
corollaries  or  exercises  instead  of  receiving  the  emphasis  which  attaches  to 
a  theorem;  in  fact,  probably  no  injury  would  result  from  a  similar  treat- 
ment of  many  of  the  theorems  stated  in  roman  t)rpe. 

The  distinction  in  emphasis  is  desirable  not  only  for  guidance  in  omitting 
theorems  in  courses  which  are  necessarily  abbreviated,  but  it  is  also  of  the 
highest  importance  in  courses  in  which  all  of  the  theorems  are  given.  An 
orderly  classification  of  theorems  in  the  student's  mind,  a  notion  of  the 
dependence  of  the  minor  theorems  on  the  more  basal  ones,  and  an  apprecia- 
tion of  their  relative  importance  are  of  the  utmost  direct  value  to  the  student 
and  furnish  him  with  the  only  possible  means  of  permanently  retaining 
geometrical  knowledge  in  usable  form.  The  direct  value  mentioned  ar;ses 
both  from  the  power  acquired  and  also  from  the  essential  grasp  of  the 
subject,  which  is  the  purpose  of  education.  It  is  a  fundamental  char- 
acteristic of  the  mind  from  which  there  is  no  escape  that  any  clear  impression 
of  a  vast  field  must  have  exactly  such  distinctions  in  emphasis  as  are 
outlined  here  for  geometry.  These  statements  and  this  arrangement  are 
intended  to  be  of  assistance  to  the  teacher  in  guiding  him  as  to  the  emphasis 
to  be  laid  upon  theorems  during  the  course  and  especially  at  the  completion 
of  a  given  book  or  chapter. 

(6)  Trigonometric  ratios. — Attention  is  called  to  the  paragraphs  under 
XIII,  2-4,  on  the  computation  of  two-place  tables  of  sines,  cosines,  and 
tangents  from  actual  measurements,  orovided  the  pressure  of  time  due  to 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS  33 

examining  bodies  is  not  too  great.  This  work  can  be  done  with  about  the 
same  amount  of  effort  that  is  expended  by  the  student  on  the  ordinary 
geometrical  theorems  of  the  same  class.  Its  importance  is  due  to  the  fact 
that  such  a  small  table  will  really  present  the  fundamental  ideas  of 
trigonometry  and  will  enable  the  student  to  solve  right  triangles  in 
the  trigonometric  sense. 

(7)  Abbreviations. — In  a  large  number  of  instances  theorems  are  stated 
in  condensed  or  abbreviated  form  and  the  statement  of  a  number  of  theorems 
is  often  combined  into  one.  This  is  done  only  for  the  purpose  of  reducing 
the  length  of  this  report.  It  is  to  be  understood  that  such  abbreviated 
statements  are  made  only  for  the  teacher  and  should  not  be  presented  to 
the  student  in  this  form.  In  particular,  it  is  probably  preferable  to  use 
words  instead  of  letters  in  statements  for  high-school  pupils  of  such  theorems 
as  those  in  II,  1-4. 

The  numbers  which  follow  each  of  the  theorems  are  references  to  a 
syllabus  prepared  by  a  committee  of  the  Association  of  Mathematics 
Teachers  of  New  England,  1906. 

(8)  Omissions. — Since  the  quantities  which  appear  in  geometry  are 
often  treated  by  means  of  their  numerical  measures,  the  introduction  of  any 
useful  algebraic  facts  is  completely  justifiable;  in  particular,  the  algebraic 
theory  of  proportion  may  well  be  employed;  but  such  algebraic  material 
is  not  included  in  this  syllabus  since  the  syllabus  deals  primarily  with 
geometric  topics. 

The  following  list  shows  the  omissions  from  the  New  England  Syllabus: 
Plane  geometry  omissions:    C2,  G12  (2d  part),  G20,  J13,  L2,  N6  (2d 

part),  N13,  Pi,  P2  (see  note  to  XV,  i),  P6,  P12,  T2,  T4,  T5. 

Solid  geometry  omissions:  E7,  E8,  Egb,  F12,  F13,  F14,  H2  (but  see  IV,  7), 

K2,  K3  (see  note  to  I,  2),  M6,  M7,  MS,  M9,  Q3,  Q4,  QSb,  Q9  (see  VI, 

note),  Ri,  R2  (see  note  to  VI,  13),  R9,  Rio,  R13,  R14  (see  note  to  VI,  22), 

S3  (see  preface,  3),  S7. 

THEOREMS  OF  PLANE  GEOMETRY 

I.     Theorems  for  Informal  Proof 

(The  following  theorems  may  be  stated  as  assumptions,  or  may  be  given 
such  informal  proof  as  the  circumstances  may  demand.) 

1.  All  straight  angles  are  equal.^     [*] 

2.  All  right  angles  are  equal.     [*] 

3.  The  sum  of  two  adjacent  angles  whose  exterior  sides  lie  in  the  same  straight  line 
equals  a  straight  angle.     [Ji.] 

4.  If  the  sum  of  two  adjacent  angles  equals  a  straight  angle  their 
exterior  sides  form  a  straight  line.     [J2.] 

» Reference  numbers  are  to  the  New  England  Syllabus.  Where  an  asterisk  [*]  replaces  the  reference 
number,  the  theorem  is  not  contained  in  that  syllabus. 


34  NATIONAL  EDUCATION  ASSOCIATION 

5.  Only  one  perpendicular  can  be  erected  from  a  given  point  in  a  given 
line.     [G3.] 

6.  The  length  of  a  circle  (circumference)  lies  between  the  lengths  of  perimeters  of  the 
inscribed  and  circumscribed  convex  polygons.     [P13.] 

(It  is  recommended  that  this  statement  be  used  as  a  definition  to  be  inserted  at 
context.) 

7.  The  area  of  a  circle  lies  between  the  areas  of  inscribed  and  circumscribed  convex 
polygons.     [P14.] 

(It  is  recommended  that  this  statement  be  used  as  a  definition  to  be  inserted  at 
context.) 

8.  Two  lines  parallel  to  the  same  line  are  parallel  to  each  other.     [*] 

9.  Vertical  angles  are  equal.     [J3.] 
(Very  informal  proof  sufl&cient.) 

10.  Complements  of  equal  angles  are  equal.     [*] 

11.  Supplements  of  equal  angles  are  equal.     [*] 

12.  The  bisectors  of  vertical  angles  lie  in  a  straight  line.     [J4.] 

13.  Any  side  of  a  triangle  is  less  than  the  sum  of  the  other  two  and 
greater  than  their  difference.     [*] 

14.  A  diameter  bisects  a  circle.     [A5.] 

15.  A  straight  line  intersects  a  circle  at  most  in  two  points.    [G6.] 

II.     Congruence  of  Triangles 

1.  Any  two  triangles^  ABC  and  A'B'C'  are  congruent  if: 
(i)  a=a'  b=b'  C=C'  [Ai.] 

(2)  a=a'            B=B'  C=C'            [A2.] 

(3)  a  =  a'            b=b'  c=c'             [A3.] 

(4)  a=a'             c=c'  C=C'=90°  [A4.] 
(State  these  in  detail  and  in  English.  See  preface,  article  7.) 

2.  A  triangle  is  determined  when  the  following  are  given:  (i)  a,  h,  C; 
(2)  a,  B,  C;   (3)  a,  h,  c;  (4)  a,  c,  C  =  9o°.     [*] 

(Synonymous  to  i.) 

3.  Construction  of  triangles  from  given  parts ;  measurement  of  unknown 
parts  by  ruler  and  protractor.  Given:  (i)  a,  6,  C;  (2)  a,  B,  C;  (3)  a,  J,  c; 
(4)  a,  c,  C,  possibly  two  solutions.     [*] 

(This  is  the  fundamental  elementary  idea  of  trigonometry.) 

4.  In  any  two  triangles  if  a=a'  and  6  =  6',  either  of  the  inequalities  c>c'  or  C>C' 
is  a  consequence  of  the  other.     [O3,  O4.I 

III.     Congruent  Right  Triangles 

I.  Two  right  triangles  are  congruent  if,  aside  from  the  right  angles, 
any  two  parts,  not  both  angles,  in  the  one  are  equal  to  corresponding 
parts  of  the  other.     [A4.] 

(Very  important  subcase  of  II,  i.) 

<  In  this  syllabus  the  angles  of  a  triangle  ABC  are  denoted  by  the  capital  letters  A.  B,  and  C;  the  sides 
are  denoted  by  small  letters  a,  b,  and  c,  where  a  !s  the  side  opposite  the  angle  A,  etc. 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS 


35 


2.  If  two  oblique  lines  c  and  c'  be  drawn  from  a  point  in  a  perpendicular 
P  to  a,  line  AA',  cutting  off  distances  d  and  d\  then 
any  one  of  the  equalities  c=c',  d=d',  A=A',  B  =  B', 
is  a  consequence  of  any  other.     [G5.] 

3.  A  diameter  perpendicular  to  a  chord  bisects  the 
chord,  the  subtended  angle  at  the  center,  and  the  sub- 
tended arc;  conversely,  a  diameter  which  bisects  a  chord 
is  perpendicular  to  it.     [Gsb,  G8.] 

(Corollary  to  2.    See  also  IV,  3.) 

4.  If  two  oblique  lines  c  and  c'  be  drawn  from  a  point  in  a  perpendicular  ^  to  a  line 
AA',  cutting  ofif  unequal  distances  d  and  d',  then  either  of  the  inequahties  c>c',  d>d',  is 
a  consequence  of  the  other.    [O5,  06.] 

(In  particular,  c  is  greater  than  p.) 

5.  If,  in  a  triangle  ABC,  a=b,  the  perpendicular  from  C  on  c  divides  the 
triangle  into  two  congruent  triangles.     [*] 

6.  In  a  triangle  ABC,  either  of  the  equations  a=b,A=B,is  a  consequence 
of  the  other.     [Gi,  G2.] 

7.  In  a  triangle  ABC,  either  of  the  statements  a>b,  A>B,  is  a  consequence  of  the 
other.     [Oi,  O2.] 


Subtended  Arcs,  Angles,  and  Chords 

I.  In  the  same  circle,  or  in  equal  circles,  any  one 
of  the  equations  d  =  d',  k=k',  c=c',  0  =  0',  is  a  con- 
sequence of  any  other  one  of  them.    [A6,  7,  8,  9,  G9.] 

2.  Any  one  of  the  inequalities  (see  figure) 
d<d\  0>0',  c>c',  k>k' 
is  a  consequence  of  any  other  one  of  them.     [O7,  8.] 

3.  In  any  circle  an  angle  at  the  center  is  measured  by  its  intercepted  arc, 

as.] 

(Only  the  commensurable  case.) 

4.  If  a  circle  is  divided  into  equal  arcs,  the  chords  of  these  arcs  form  a 
regular  polygon.     [G12,  last  part.] 

5.  To  construct  an  angle  equal  to  a  given  angle.     [J14.] 

(Regular  polygons  may  be  constructed  approximately  by  means  of  a  protractor.  In 
the  same  way  other  approximate  constructions  may  be  introduced  which  depend  upon 
the  protractor.) 


V.     Perpendicular  Bisectors 

1.  The  perpendicular  bisector  of  a  line-segment  is  the  locus  of  points 
equidistant  from  the  ends  of  the  segment.     [Si.] 

2.  To  draw  the  perpendicular  bisector  of  a  given  line-segment.     [G14.] 

3.  To  erect  a  perpendicular  at  a  given  point  in  a  line,     [*] 
(Corollary  to  2.) 


36  NATIONAL  EDUCATION  ASSOCIATION 

4.  To  drop  a  perpendicular  from  a  given  point  to  a  given  line.     [D5.] 
(Corollary  to  2.) 

5.  To  bisect  a  given  arc  or  angle.     [G15,  16.] 
(See  III,  3.) 

6.  To  inscribe  a  square  in  a  circle.     [G18.] 

7.  One  and  only  one  circle  can  be  circumscribed  about  any  triangle.     [G13.] 

8.  Three  points  determine  a  circle.  Two  circles  can  intersect,  at  most,  in  two  points; 
this  will  happen  when  the  distance  between  their  centers  is  less  than  the  sum  of  the  radii 
and  greater  than  the  difference  of  the  radii.    [G7.] 

(Corollary  to  7.) 

9.  Given  an  arc  of  a  circle,  to  find  its  center.     [*] 
(Corollary  to  7.) 

10.  A  circle  may  be  circumscribed  about  any  regular  polygon.    [G13,  third  part.] 

11.  The  perpendicular  bisectors  of  the  sides  of  a  triangle  meet  in  a  point.     [T3.] 


VI.    Bisectors  of  Angles 

1.  The  bisector  of  any  angle  is  the  locus  of  points  equidistant  from  the 
sides  of  the  angle.     [S2.] 

2.  A  circle  can  be  inscribed  in  any  triangle.     [G13,  second  part.] 
(Construction  to  be  given.) 

3.  A  circle  can  be  inscribed  in  any  regular  polygon.     [G13,  last  part.) 

4.  Of  the  inscribed  and  circumscribed  regular  polygons  of  n  and  2w  sides  for  a  given 
circle,  to  draw  the  remaining  three  polygons  when  one  is  given.     [G17.] 

5.  The  bisectors  of  the  angles  of  any  triangle  meet  in  a  point.     [Ti.] 
(Corollary  to  2.) 

y  VII.     Parallels 

1.  When  two  lines  are  cut  by  a  transversal  the  alternate  interior  angles 
are  equal  if,  and  only  if,  those  two  lines  are  parallel.     [Half  of  Di,  2.] 

When  two  lines  are  cut  by  a  transversal,  the  alternate  interior  angles  are  unequal  if, 
and  only  if,  the  lines  are  not  parallel. 
(Synonymous  to  i.) 

2.  When  two  lines  are  cut  by  a  transversal  the  corresponding  angles  are  equal,  and 
the  two  interior  angles  on  the  same  side  of  the  transversal  are  supplementary  if,  and  only 
if,  the  two  lines  are  parallel.     [Half  of  Di,  2.] 

(Corollary  to  i.) 

3.  Two  lines  in  the  same  plane  perpendicular  to  the  same  line  are 
parallel.     [D4,  G4.] 

(Only  one  perpendicular  can  be  let  fall  from  a  point  without  a  line  to  that  line. 
Synonymous  to  3.) 

4.  A  line  perpendicular  to  one  of  two  parallels  is  perpendicular  to  the 
other  also.     [D3.] 

(Corollary  to  i.) 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS  37 

5.  If  two  angles  have  their  sides  respectively  parallel  or  respectively  perpen- 
dicular to  each  other,  they  are  either  equal  or  supplementary.     [J7.] 

6.  Thru  a  given  point  to  draw  a  straight  line  parallel  to  a  given  straight 
line.     [D6.] 

7.  A  parallelogram  is  divided  into  two  congruent  triangles  by  either 
diagonal.     [*] 

8.  In  any  parallelogram  the  opposite  sides  are  equal,  the  opposite 
angles  are  equal,  the  diagonals  bisect  each  other.     [D7.] 

(Corollary  to  7.) 

9.  In  any  convex  quadrilateral,  if  the  opposite  sides  are  equal,  or  if  the 
opposite  angles  are  equal,  or  if  one  pair  of  opposite  sides  are  equal  and 
parallel,  or  if  the  diagonals  bisect  each  other,  the  figure  is  a  parallelogram. 
[D8.] 

VIII.    Angles  of  a  Triangle 

1.  In  any  triangle  the  sum  of  the  angles  is  two  right  angles.     [J5(b).] 

2.  In  any  triangle,  any  exterior  angle  is  equal  to  the  sum  of  the  two 
opposite  interior  angles.     [J5(a).] 

(Synonymous  to  i.) 

3.  The  sum  of  the  interior  angles  of  a  polygon  of  n  sides  is  2  (w — 2)  right  angles. 
[J6.] 

4.  To  inscribe  a  regular  hexagon  in  a  circle.     [G19.] 
(To  construct  an  angle  of  60°.     Synonymous  to  4.) 


IX.    Inscribed  Angles 

1.  An  angle  inscribed  in  a  circle  is  measured  by  half  of  its  intercepted 
arc.     [J9.] 

2.  Angles  inscribed  in  the  same  segment  are  equal  to  each  other.     [*] 

3.  An  angle  inscribed  in  a  semicircle  is  a  right  angle.     [*] 

4.  The  two  arcs  intercepted  by  parallel  secants  are  equal.     [Gii.] 

5.  The  angle  between  a  tangent  and  a  chord  is  measured  by  half  the 
intercepted  arc.     [Jio.] 

6.  The  angle  between  any  two  lines  is  measured  by  half  the  sum,  or  half 
the  difference,  of  the  two  arcs  which  they  intercept  on  any  circle,  according 
as  their  point  of  intersection  lies  inside  of,  or  outside  of,  the  circle.    [Ji  i,  12.] 

7.  The  tangent  to  a  circle  at  a  given  point  is  perpendicular  to  the  radius  at 
that  point.     [Li,  3.] 

8.  For  a  given  chord,  to  construct  a  segment  of  a  circle  in  which  a  given  angle  can 
be  inscribed.     [J15.] 

9.  To  draw  a  tangent  to  a  given  circle  thru  a  given  point.     [L4.] 

10.  The  tangents  to  a  circle  from  an  external  point  are  equal.     [Gic] 
(Corollary  to  7.) 


38  NATIONAL  EDUCATION  ASSOCIATION 

X.     Segments  Made  by  Parallels 

1.  If  a  series  of  parallel  lines  cut  off  equal  segments  on  one  transversal, 
they  cut  off  equal  segments  on  any  other  transversal.     [D9.] 

2.  The  segments  cut  of  on  two  transversals  by  a  series  of  parallels  are 
proportional     [See  Nio.] 

(Only  the  commensurable  case.) 

3.  A  line  divides  two  sides  of  a  triangle  proportionally,  the  segments  of  the 
two  sides  being  taken  in  the  same  order,  if,  and  only  if,  it  is  parallel  to  the 
third  side.     [Ni,  2.] 

(Only  the  commensurable  case.) 

4.  To  divide  a  line-segment  into  n  equal  parts  or  into  parts  proportional 
to  any  given  segments.     [N9,  10.] 

5.  To  find  a  fourth  proportional  to  three  given  line-segments.    [Nil] 

XI.  Similar  Triangles 

1.  Two  triangles  ABC  and  A'B'C'  are  similar  if 

(i)  A=A'  B=B'  C=C'  [N3.] 

or  (2)  a=ka'  b=kb'  C=C'  [N4.] 

or  (3)  a=ka'  b=kb'  c=kc'  [N5.] 

where  k  is  a  constant  factor  of  proportionality. 
(See  preface,  article  7.) 

2.  Given  a  fixed  point  P  and  a  circle  C,  the  product  of  the  two  distances 
measured  along  any  straight  line  thru  B^from  P  to  the  points  of  intersection  with 
C,  is  constant.  This  product  is  also  equal  to  the  square  of  the  tangent  from  P 
to  C  if  P  is  an  external  point.     [N18.] 

3.  The  bisector  of  any  angle  of  a  triangle  divides  the  opposite  side  into 
segments  proportional  to  the  adjacent  sides.     [Half  of  N6.] 

4.  To  construct  a  triangle  similar  to  a  given  triangle.     [*] 

(Drawing  triangles  to  scale;  measurements  of  remaining  parts  to  scale.  Basal  in 
trigonometry.) 

XII.  Similar  Figures 

1.  Polygons  are  similar  if,  and  only  if,  they  can  be  decomposed  into  triangles 
which  are  similar  and  similarly  placed.     [N7,  8.] 

2.  Regular  polygons  of  the  same  number  of  sides  are  similar.    [N14.] 

3.  The  perimeters  of  similar  polygons  are  proportional  to  any  two 
corresponding  lines  of  the  polygons.     [N15.] 

4.  The  circumferences  of  any  two  circles  are  proportional  to  their 
diameters,  thus  c  =  27rr,  where  ir  is  constant.     [Pi5«] 

(w  =  3 .  14 to  be  computed  later.) 

5.  To  construct  a  polygon  similar  to  a  given  polygon.     [*] 
(Drawings  to  scale;  maps,  house  plans;  readings  from  drawings;  plotting  of  measure- 
ments.   Essential  in  surveying.) 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS  39 

XIII.     Similar  Right  Triangles 

(The  committee  feels  that  numbers  2,  3,  4  following  should  have  a  place  where  time 
for  their  discussion  can  be  secured,  which  will  doubtless  be  the  case  except  under  pressure 
from  examining  bodies.) 

1.  Any  two  right  triangles  are  similar  if  an  acute  angle  of  the  one  is 
equal  to  an  acute  angle  of  the  other,  or  if  any  two 
sides  of  one  are  proportional  to  the  corresponding 
sides  of  the  other.     [*] 

2.  For  a  given  acute  angle  A,  the  sides  of  any  right 
triangle  ABC  (C=go°)  form  fixed  ratios,  called  the  sine 
(a/c),  the  cosine  (b/c),  the  tangent  (a/b).     [*] 

3.  Computation  of  a  two-place  table  of  sines,  cosines,  tangents  from 
actual  measurements.     [*] 

(Probably  a  two-place  table  for  every  5°  or  10°;  to  be  done  by  students,  preferably 
on  squared  paper.) 

4.  Solution  of  right  triangles  with  given  parts  by  use  of  the  preceding  table  of 
ratios.    [*] 

(Height  and  distance  exercises.) 

XIV.     Right  Triangles 
I.  In  any  right  triangle  ABC  the  perpendicular  let  fall  from  the  right 
angle  upon  the  hypotenuse  divides  the  triangle  into  similar  right  triangles, 
each  similar  to  the  original  triangle.     ['*'] 

2.  The  length  of  the  perpendicular  p  is  the  mean 
proportional  between  the  segments  m  and  n  of  the 
hypotenuse;  i.e., />^= WW.     [P8.] 

3.  Either  side,  a  or  b,  is  the  mean  proportional 
between  the  whole  hypotenuse  c  and  the  adjacent  segment  m  ov  n;  that  is, 
a*=cm;   b^=cn.     [Pp.] 

4.  To  fin.d  a  mean  proportional  between  two  given  line-segments. 
[N12.] 

5.  The  sum  of  the  squares  of  the  two  sides  of  a  right  triangle  is  equal 
to  the  square  of  the  hypotenuse;  a^+b'=c'.     [Pic] 

(It  should  be  noticed  that  the  proposition  can  be  proved  either  algebraically  or 
geometrically.) 

6.  In  any  triangle  ABC,  if  B  is  less  than  90°,  then  b''  =  a'+c'—2cm',  if  B  is  greater 
than  90",  then  b^  =  a^+c^-\-2cm,  where  m  is  the  projection  of  a  on  c.    [P17.] 

(See  figure  under  2.) 

7.  Given  the  radius  of  a  circle  and  a  perimeter  of  an  inscribed  regular  polygon  of 
n  sides,  to  find  the  perimeter  of  the  circumscribed  regular  polygon  of  n  sides  and  the 
perimeter  of  the  inscribed  regular  polygon  of  2»  sides. 

[G17.    See  also  X4.] 

8.  To  calculate  tt  approximately.     [*] 

XV.     Areas 
I.  The  area  of  a  rectangle  is  the  product  of  its  base  and  its  altitude; 
i.e.,  a=bh.     [Pi,  2,  3.] 

(This  formula  may  be  taken  as  the  definition  of  area.) 


40  NATIONAL  EDUCATION  ASSOCIATION 

2.  Parallelograms  or  triangles  of  equal  bases  and  altitudes  are  equivalent. 

[Ci.]     - 

3.  The  area  of  a  parallelogram  is  the  product  of  its  base  and  its  altitude ; 

i.e.,  a  =  bh.     [P4.] 

4.  The  area  of  a  triangle  is  one-half  the  product  of  its  base  and  its 
altitude;  i.e.,  a  =  ibh.     [P5.] 

5.  The  area  of  a  trapezoid  is  one-half  the  product  of  its  altitude  and 
the  sum  of  its  bases;  i.e.,  o  =  J  {hi-\-h2)h.     [P7.] 

6.  The  areas  of  similar  triangles  or  polygons  are  proportional  to  the 
squares  of  corresponding  lines.     [N16,  17.] 

7.  The  area  of  a  regular  polygon  is  one-half  the  product  of  its  perimeter 
and  its  apothem.     [Pii.] 

8.  The  area  of  any  circle  is  one-half  the  product  of  its  circumference 
and  its  radius;  i.e.,  a  =  7rr^     [P16.] 

9.  The  areas  of  two  circles  are  proportional  to  the  squares  of  their  radii.     [*] 
(May  be  treated  as  suggested  in  preface,  article  3.) 

10.  To  construct  a  square  equivalent  to  the  sum  of  two  given  squares.     [*] 
(Pythagorean  proposition.) 

11.  To  construct  a  square  equivalent  to  a  given  rectangle.     [C3.] 
(Mean  proportional.    See  X6.) 

THEOREMS  OF  SOLID  GEOMETRY 

In  this  part  the  same  general  principles  apply  as  were  stated  in  the 
preface  above. 

Thruout,  but  particularly  in  divisions  I  and  II  below,  very  great 
emphasis  should  be  laid  upon  the  student's  real  grasp  of  the  conceptions,  of 
the  space  figures,  and  of  the  significance  of  the  theorems.  While  the  theo- 
rems in  division  I  will  be  seen  to  need  little  or  no  suggestion  of  proof,  it  is 
a  mistake  to  suppose  that  they  can  be  hastened  over;  on  the  contrary,  even 
in  these,  the  teacher  should  spare  no  pains  to  make  sure  that  the  student's 
mental  picture  is  quite  vivid,  resorting  to  formal  proof  when  necessary. 
To  this  end,  illustrations,  figures,  models,  various  forms  of  presentation, 
and  all  such  aids  are  legitimate  thruout  the  course  in  solid  geometry. 

I.    Theorems  for  Informal  Proof 

1.  If  two  planes  cut  each  other,  theu-  intersection  is  a  straight  line.   [S4.] 

2.  Two  dihedral  angles  have  the  same  ratio  as  their  plane  angles.     [K2, 

3,4-] 

(Equivalent  to  K3.) 

3.  Every  section  of  a  cone  made  by  a  plane  passing  thru  its  vertex  is 
a  triangle.     [M4.] 

4.  Every  section  of  a  cylinder  made  by  a  plane  passing  thru  an  element 
is  a  parallelogram.     [M2.] 

5.  The  area  of  a  sphere  lies  between  the  areas  of  circumscribed  and  inscribed  convex 
polyhedrons.     [*] 

(It  is  recommended  that  this  statement  be  used  as  a  definition  to  be  inserted  at 
context.) 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS  41 

6.  The  volume  of  a  sphere  lies  between  the  volumes  of  circumscribed  and  inscribed 
convex  polyhedrons.     [*] 

(It  is  recommended  that  this  statement  be  used  as  a  definition  to  be  inserted  at 
context.) 

7.  The  projection  of  a  straight  line  upon  a  plane  is  a  straight  line.     [S8.] 

II.     Corollaries  from  Plane  Geometry 

(The  ability  to  make  the  transfer  from  plane  geometry  to  solid  geometry, 
and  vice  versa,  in  forming  conceptions  and  in  logical  deductions  is  of  the 
utmost  importance.  The  following  theorems  are  easily  reducible  to  plane 
geometry  in  at  most  two  or  three  planes.  The  intention  is  that  careful 
proofs  be  given,  but  the  student  should  see  that  these  theorems  result 
immediately  from  known  theorems  of  plane  geometry.) 

1.  The  intersections  of  two  parallel  planes  with  any  third  plane  are 
parallel.     [Fi.] 

2.  A  plane  containing  one  and  only  one  of  two  parallel  lines  is  parallel 
to  the  other.     [F7.] 

3.  If  a  straight  line  is  parallel  to  a  plane,  the  intersection  of  the  plane 
with  any  plane  drawn  thru  the  line  is  parallel  to  the  line.     [*] 

4.  Thru  a  given  point  only  one  plane  can  be  passed  parallel  to  two 
straight  lines  not  in  the  same  plane.     [Fic] 

(Derived  from  2.) 

5.  Thru  a  given  straight  line  only  one  plane  can  be  passed  parallel  to 
any  other  given  straight  line  in  space,  not  parallel  to  the  first.     [Fii.] 

(Derived  from  2,) 

6.  Thru  a  given  point  only  one  plane  can  be  drawn  parallel  to  a  given 
plane.     [Fp.] 

(Synonymous  to  4.) 

7.  If  a  perpendicular  PO  be  let  fall  from  a  point  P  to  a  plane  L,  any  one 
of  the  equalities 

a^=a',c=c',B  =  B',A=A' 
p  is  a  consequence  of  any  other  of  them;   and  any  one 

is.  of  the  inequalities 

r?-— -^S.        j  a>a\c>c',  B<  B',  A>A' 

/  qUS — & — Bfci.   /        jg  g^  consequence  of  any  other  one  of  them.     [Oio, 
^ '         H7.     See  also  S8.] 

8.  The  perpendicular  PO  is  shorter  than  any  otlique  line.     [O9.] 

9.  Two  straight  lines  are  parallel  to  each  other  if,  and  only  if,  they  are' 
both  perpendicular  to  some  one  plane.     [F2,  3.] 

10.  If  two  straight  lines  are  parallel  to  a  third,  they  are  parallel  to 
each  other.     [F4.] 

(Derived  from  9.) 

11.  Two  planes  are  parallel  to  each  other  if,  and  only  if,  they  are  both 
perpendicular  to  some  one  straight  line.     [F5,  6.] 

(Derived  from  9.) 


42  NATIONAL  EDUCATION  ASSOCIATION 

12.  The  locus  of  points  equidistant  from  the  extremities  of  a  straight 
line  is  a  plane  perpendicular  to  that  line  at  its  middle  point.     [S5.] 

13.  If  two  straight  lines  are  cut  by  three  parallel  planes,  their  corre- 
sponding segments  are  proportional.     [See  Mi.] 

14.  The  locus  of  points  equidistant  from  two  intersecting  planes  is  the 
figure  formed  by  the  bisecting  planes  of  their  dihedral  angles.     [S6.] 

III.     Planes  and  Lines 

1.  If  a  straight  line  is  perpendicular  to  each  of  two  other  straight  lines 
at  their  point  of  intersection,  it  is  perpendicular  to  every  line  in  their  plane 
thru  the  foot  of  the  perpendicular.     [Ei.] 

2.  Every  perpendicular  that  can  be  drawn  to  a  straight  line  at  a  given 
point  lies  in  a  plane  perpendicular  to  the  line  at  the  given  point.     [E2.] 

(Corollary  to  i.) 

3^  Thru  any  point  only  one  plane  can  be  drawn  perpendicular  to  a 
given  line.     [E5.] 

(Corollary  to  i,  and  II,  11.) 

4.  Thru  a  given  point  only  one  perpendicular  can  be  drawn  to  a  given 
plane.     [E6.] 

(Corollary  to  i.) 

5.  If  two  angles  have  their  sides  respectively  parallel  and  lying  in  the 
same  direction,  they  are  equal,  and  their  planes  are  parallel.     [Ki,  F8.] 

6.  //  a  line  meets  Us  projection  on  a  plane,  any  line  of  the  plane  perpen- 
dicidar  to  one  of  them  at  their  intersection  is  perpendicular  to  the  other  also.     [*] 

7.  Between  any  two  straight  lines  not  in  the  same  plane,  one  and  only  one  common 
perpendicular  can  be  drawn,  and  this  common  perpendicular  is  the  shortest  line  that  can 
be  drawn  between  the  two  lines.    [E12.I 

8.  Two  planes  are  perpendicular  to  each  other  if,  and  only  if,  a  line 
perpendicular  to  one  of  them  at  a  point  in  their  intersection  lies  in  the  other. 

[E3,  4.] 

9.  If  a  straight  line  is  perpendicular  to  a  plane,  every  plane  passed  thru  the  line  is 
perpendicular  to  the  first  plane.    [Epa.] 

(Corollary  to  8.) 

10.  If  two  intersecting  planes  are  each  perpendicular  to  a  third  plane,  their  inter- 
section is  also  perpendicular  to  that  plane.     [Exo.] 

(Corollary  to  8.) 

11.  Thru  a  given  straight  line  oblique  to  a  plane ,  one  and  only  one  plane 
can  he  passed  perpendicular  to  the  given  plane.     [Eii.] 

12.  The  acute  angle  which  a  straight  line  makes  with  its  own  projection 
on  a  plane  is  the  least  angle  which  it  makes  with  any  line  of  the  plane. 

[013.] 

13.  Two  right  prisms  are  congruent  if  they  have  congruent  bases  and 
equal  altitudes.     [Bi.] 

14.  //  parallel  planes  cut  all  the  lateral  edges  of  a  pyramid,  or  a  prism,  the 
sections  are  similar  polygons;   in  a  prism,  the  sections  are  congruent;  in  a 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS  43 

Pyramid f  their  areas  are  proportional  to  the  squares  of  their  distances  from  the 
vertex.     [Mi.] 
(See  II,  13.) 

15.  Every  section  of  a  circular  cone  made  by  a  plane  parallel  to  its  base  is  a  circle,  the 
center  of  which  is  the  intersection  of  the  plane  with  the  axis.     [M5.] 

16.  Parallel  sections  of  a  cylindrical  surface  are  congruent.     [M3.] 

IV.     Spheres 

1.  Every  section  of  a  sphere  made  by  a  plane  is  a  circle.     [Mio.] 
(Several  corollaries  may  be  added.) 

2.  The  intersection  of  two  spheres  is  a  circle  whose  axis  is  the  line  of  centers.    [H4.] 

3.  The  shortest  path  on  a  sphere  between  any  two  points  on  it  is  the  minor 
arc  of  the  great  circle  which  joins  them.     [O14.] 

4.  A  plane  is  tangent  to  a  sphere  if,  and  only  if,  it  is  perpendicular  to  a 
radius  at  its  extremity.     [Mii,  12,  13.] 

5.  A  straight  line  tangent  to  a  circle  of  a  sphere  lies  in  a  plane  tangent  to  the  sphere 
at  the  point  of  contact.     [*] 

6.  The  distances  of  all  points  of  a  circle  on  a  sphere  from  either  of  its 
poles  are  equal.     [Hi.] 

7.  A  point  on  the  surface  of  a  sphere,  which  is  at  the  distance  of  a 
quadrant  from  each  of  two  other  points,  not  the  extremities  of  a  diameter, 
is  the  pole  of  the  great  circle  passing  thru  these  points.     [H3.] 

8.  A  sphere  can  be  inscribed  in  or  circumscribed  about  any  given  tetrahedron.    [H5.] 

9.  A  spherical  angle  is  measured  hy  the  arc  of  a  great  circle  described  from 
its  vertex  as  a  pole  and  included  between  its  sides  {produced  if  necessary). 
[K5.] 

V.     spherical  Triangles  and  Polygons 

(Every  theorem  stated  here  may  also  be  stated  as  a  theorem  on  polyhedral  angles.) 

1.  Each  side  of  a  spherical  triangle  is  less  than  the  sum  of  the  other 
two  sides.     [On  (b).     See  also  (a).] 

2.  The  sum  of  the  sides  of  a  spherical  polygon  is  less  than  360°. 
[O12  (b).     See  also  (a).] 

3.  The  sum  of  the  angles  of  a  spherical  triangle  is  greater  than  180°  and  less  than 
54o^    [K8.J 

4.  If  A'B'C  is  {he  polar  triangle  of  ABC ^  then j  reciprocally,  ABC  is  the 
polar  of  A'B'C.     [K6.] 

5.  In  two  polar  triangles  each  angle  of  the  one  is  the  supplement  of  the 
opposite  side  in  the  other.     [K7.] 

6.  Vertical  spherical  triangles  are  symmetrical  and  equivalent.     [C8,  H6.] 

7.  Two  triangles^  on  the  same  sphere  are  either  congruent  or  sym- 
metrical if 

c=c'  [B2  (b).     See  also  (a).] 

C=C'  [B3  (b).     See  also  (a).] 

C=C'  [B4  (b).     See  also  (a).] 

C=C'  [B5  (b).     See  also  (a).] 

8.  Either  of  the  equations  a  =  &,  A  =  B  is  a  consequence  of  the  other.    [*] 
•  The  same  notation  is  used  as  in  the  plane  triangles. 


a=a' 

b=b' 

or  a=a' 

b=b' 

or  a=a' 

B=B' 

orA=A' 

B=B' 

44  NATIONAL  EDUCATION  ASSOCIATION 


VI.     Mensuration 

(The  relation  between  the  areas  and  volumes  of  similar  solids  may  be  treated  as 
corollaries  in  individual  cases.  See  preface,  article  3.  It  is  understood  that  certain 
statements  concerning  limits  may  be  assumed  either  explicitly  or  implicitly;  these  are 
not  stated  as  theorems.    See  Q3,  4,  9,  R9,  10.) 

1.  An  oblique  prism  is  equivalent  to  a  right  prism  whose  base  is  a  right 
section  of  the  oblique  prism  and  whose  altitude  is  a  lateral  edge  of  the 
oblique  prism.     [C4.] 

2.  A  plane  passed  thru  two  diagonally  opposite  edges  of  a  parallelopiped 
divides  it  into  two  equivalent  triangular  prisms.     [C6.] 

3  The  lateral  area  of  a  prism  is  the  product  of  a  lateral  edge  and  the 
perimeter  of  a  right  section.     [Qi.] 
(Corollary  of  plane  geometry.) 

4.  The  lateral  area  of  a  regular  pyramid  is  one-half  the  product  of  the  slant 
height  and  the  perimeter  of  the  base.     [Q2.] 

(Corollary  of  plane  geometry.) 

5.  The  lateral  area  of  a  right  circular  cylinder  is  the  product  of  the  altitude 
and  the  circumference  of  the  base;  i.e.,  s=2irrh.     [Q5.] 

6.  The  lateral  area  of  a  right  circular  cone  is  one-half  the  product  of  the 
slant  height  and  the  circumference  of  the  base;  i.e.,  s=Trrl.     [Q6.] 

7.  The  lateral  area  of  a  frustum  of  a  regular  pyramid  is  one-half  the  product  of  the 
slant  height  and  the  sum  of  the  perimeters  of  the  bases.     [Q7.] 

8.  The  lateral  area  of  a  frustum  of  a  right  circular  cone  is  one-half  the  product  of  the 
slant  height  and  the  sum  of  the  circumferences  of  the  bases.     [Q8  (a).] 

9.  The  area  of  a  zone  is  the  product  of  its  altitude  and  the  circumference 
of  a  great  circle;  i.e.,  s=2irrh.     [Qic] 

(Lemma  for  10  below.) 

10.  The  area  of  a  sphere  is  the  product  of  its  diameter  and  the  cir- 
cumference of  a  great  circle;  i.e.,  s=47rr^.     [Qn.] 

11.  The  area  of  a  lune  is  to  the  surface  of  a  sphere  as  the  angle  of  the  lune  is  to 
360^    [Q12.] 

12.  The  area  of  a  spherical  triangle  is  to  the  area  of  the  sphere  as  its  spherical  excess 
is  to  720**.     [Q13.] 

13.  The  volume  of  a  rectangular  parallelopiped  is  the  product  of  its 
three  dimensions.     [Ri,  2,  3.] 

(This  may  be  taken  as  a  definition.) 

14.  The  volume  of  any  parallelopiped  is  the  product  of  its  base  and 
altitude.     [C5,  R4.] 

15.  The  volume  of  any  prism  is  the  product  of  its  base  and  its  altitude. 
[R5,6.] 

16.  The  volume  of  any  pyramid  is  one-third  the  product  of  its  base  and  its 
altitude.     [C7,  R7,  8,] 

17.  The  volume  of  a  circular  cylinder  is  the  product  of  its  base  and  its 
altitude,  i.e.,  v  =  Trr^h.     [Rii.] 

18.  The  volume  of  a  circular  cone  is  one-third  the  product  of  its  base  and 
its  altitude;  i.e.,  v=\Trr^h.     [R12.] 


FINAL  REPORT  ON  GEOMETRY  SYLLABUS  45 

19.  The  volume  of  a  spherical  sector  is  one- third  the  product  of  the 
radius  and  the  zone  which  is  its  base:  i.e.,  v  =  f7rr^h.     [R15.] 

20.  The  volume  of  a  sphere  is  one-third  the  product  of  its  radius  and 

its  area;  i.e.,  v  =  V3'^r^-     [R16.] 

(The  wording  suggests  a  proof,  but  that  proof  is  by  no  means  prescribed.  The 
wording  is  convenient,  the  proof  may  even  preferably  follow  21  below.) 

The  follov^ring  two  theorems,  while  not  thought  by  the  committee  to  be 
indispensable,  offer  for  both  student  and  teacher  an  outlook  for  that  larger 
view  of  geometry  and  of  mathematics  as  a  whole  which  is  very  desirable. 
They  forecast  important  principles  in  future  mathematical  courses;  they 
are  capable  of  the  most  practical  direct  applications;  they  offer  a  possi- 
bility of  organizing  and  retaining  the  important  mensuration  formulas 
given  above. 

21.7/  two  solids  contained  between  the  same  parallel  planes  are  such  that 
their  sections  by  a  plane  parallel  to  those  planes  are  equal  in  area,  the  two  solids 
have  the  same  volume.     [*] 

("  Cavalieri's  Theorem."     Formal  proof  should  not  be  given.) 

22.  The  volume  of  any  sphere,  cone,  cylinder,  pyramid,  or  prism,  or  of  any  fciistum 
of  one  of  these  solids  intercepted  by  two  parallel  planes,  is  given  by  the  formula 
v=ih{t-\-4m-{-b),  where  i  is  the  area  of  the  upper  base,  b  that  of  the  lower  base,  m  that  of 
a  base  midway  between  the  two,  and  where  h  is  the  perpendicular  distance  between  the 
two  parallel  planes.    [See  R13,  14.] 

(This  formula  also  applies  to  any  so-called  prismatoid;  it  is  conveniently  useful  in 
practical  affairs.  It  should  not  be  proved  for  the  general  case,  but  each  separate  solid 
mentioned  above,  numbers  i  to  20,  can  be  shown  to  conform  to  this  rule,  by  a  direct 
check.) 

MAXIMUM  AND  MINIMUM  LISTS 

The  committee  feels  that  it  would  be  impossible  to  set  up  a  minimum 
list,  or  a  maximum  list,  of  theorems  for  college-entrance  examinations  or  for 
other  broad  purposes,  which  would  meet  with  any  widespread  approval; 
and  that  the  influence  of  such  a  list,  if  it  were  generally  accepted,  would  be 
pernicious  in  leading  to  a  total  disregard  of  many  minor  facts  of  geometry 
which  deserve  at  least  passing  notice. 

Recognizing,  however,  a  practical  demand  for  some  criterion  on  the  part 
of  the  college  examiner,  as  well  as  on  the  part  of  the  teacher,  the  committee 
makes  the  following  recommendations  for  the  guidance  of  examiners 
and  teachers: 

(a)  All  theorems  in  black-face  type  in  this  syllabus  should  be  thoroly 
known.  The  student  should  be  able  to  state  each  of  these  upon  a  clear 
suggestion  of  its  topic;  and  to  demonstrate  each  of  them  without  hesitation 
in  accordance  with  some  definite  logical  order.  He  should  know  why  they 
are  important:  in  particular,  he  should  be  ready  to  mention  other  theorems 
in  geometry  closely  connected  with  them  and  he  should  know  any  important 
concrete  applications  of  these  theorems  in  ordinary  life. 

(b)  All  theorems  in  italics  should  be  known  to  the  student  substantially 
in  the  form  here  given,  but  some  latitude  may  be  allowed  in  combining 


46  NATIONAL  EDUCATION  ASSOCIATION 

parts  of  these  with  parts  of  other  theorems.  The  student  should  be  able 
to  prove  each  of  these  theorems  on  fairly  short  notice  and  he  should  have  a 
reasonable  idea  of  the  importance  of  each  of  them. 

(c)  The  theorems  printed  in  ordinary  roman  type  should  be  familiar  to 
the  student  when  they  are  stated  by  the  examiner,  and  the  student  should 
be  able  to  make  a  proof  for  any  one  of  them  if  allowed  a  reasonable  interval 
for  thought. 

{d)  The  theorems  printed  in  small  type,  and  indeed  many  other  facts 
of  geometry  not  given  in  the  syllabus,  may  be  used  by  the  examiner  with 
the  understanding  that  they  are  to  be  regarded  in  examinations  as  of  the 
nature  of  exercises  rather  than  as  theorems  with  which  the  student  is 
supposed  already  to  have  considerable  familiarity. 

{e)  However,  the  committee  would  suggest  (i)  that  examination  ques- 
tions involving  the  trigonometric  ratios,  XIII,  2,  3,  4,  p.  39,  be  accom- 
panied by  alternative  questions  on  other  topics,  since  some  schools  may  not 
find  time  for  these  applications;  and  (2)  that  no  questions  be  given  by 
examiners  involving  proofs  of  theorems  which  may  properly  be  taken  as 
assumptions  or  as  definitions,  such,  for  instance,  as  I,  6,  7,  p.  34;  XV,  i, 
p.  39;  VI,  13,  2i,pp.  44,  45. 

CONCLUSION 

It  should  be  said  that  the  members  of  the  committee  are  not  entirely 
agreed  as  to  certain  minor  details  of  this  report.  For  example,  some  would 
place  among  the  exercises  certain  propositions  now  in  small  type;  others 
would  prefer  to  put  some  theorems  in  black-face  type  which  are  now  in 
italics;  others  would  prefer  three  types  of  propositions  instead  of  four;  and 
some  would  modify  certain  postulates  and  would  consider  as  postulates  or 
as  propositions  to  be  demonstrated  certain  theorems  included  in  the  list  of 
those  requiring  only  informal  proof.  The  committee  does  not  regard  these 
minor  matters  of  any  great  consequence,  and  therefore  wishes  to  be  con- 
sidered as  approving  the  spirit  and  general  tenor  of  the  report,  rather  than 
as  giving  individual  sanction  to  all  such  details. 

Herbert  E.  Slaught,  Chairman 

The  University  of  Chicago,  Chicago,  111. 

William  Betz  Earle  R.  Hedrick 

East  High  School,  Rochester,  N.Y.  University  of  Missouri,  Columbia,  Mo. 

Edward  L.  Brown  Frederick  E.  Newton 

North  High  School,  Denver,  Colo.  Phillips  Academy,  Andover,  Mass. 

Charles  L.  Bouton  Henry  L.  Rietz 

Harv'ard  University,  Cambridge,  Mass.  University  of  Illinois,  Urbana,  111. 

Florian  Cajori  Robert  L.  Short 

Colorado  College,  Colorado  Springs,  Colo.  Technical  High  School,  Cleveland,  Ohio 

William  Fuller  David  Eugene  Smith 

Mechanic  Arts  High  School,  Boston,  Mass.  Teachers  College,  Columbia  University, 

New  York  City,  N.Y. 

Walter  W.  Hart  Eugene  R.  Smith 

University  of  Wisconsin,  Madison,  Wis.  The  Park  School,  Baltimore,  Md. 

Herbert  E.  Hawkes  Mabel  Sykes 

Columbia  University,  New  York  City,  N.Y.        Bowen  High  School,  Chicago,  111. 


COM M EN 2 S  ON  THE  REPORT  47 

It  is  desired  that  this  F'nal  Report  may  have  a  wide  circulation  among 
teachers  of  geometry  and  mathematicians  in  general. 

Bound  copies  of  the  report  may  be  secured  gratis  upon  application 
to  the  Commissioner  of  Education,  Department  of  the  Interior,  Washing- 
ton, D.C. 


COMMENTS  ON  THE  REPORT 

EARLE   R.   HEDRICK,   PROFESSOR   OF   MATHEMATICS,   UNIVERSITY   OF  MISSOURI, 

COLUMBIA,   MO. 

The  Committee  of  Fifteen,  whose  report  I  have  the  honor  to  present  for  your  con- 
sideration, has  had  a  history  which  is  perhaps  unique.  The  committee  was  appointed 
formally  at  the  meeting  of  the  Association  at  Denver  in  1909;  its  labors  have  been  prac- 
tically continuous  since  that  time;  its  membership  has  remained  intact;  the  report 
itself  has  received  the  widest  discussion  and  distribution,  not  alone  among  the  members 
of  the  committee,  but  rather  thruout  the  entire  membership  of  this  great  Association 
and  the  other  societies  which  have  interested  themselves  in  it.  It  now  appears  for  the 
second  time,  after  a  preliminary  presentation  by  the  chairman  of  the  committee  in  San 
Francisco  last  year,  amended  in  certain  details  to  meet  the  uniformly  friendly  suggestions 
made  to  the  committee  by  individuals  and  by  societies.  It  brings  with  it  the  unanimous 
support  of  every  member  of  the  committee,  the  approval  of  individuals  and  societies 
thruout  the  country,  the  tentative  indorsement  of  this  Association  of  the  preliminary 
draft  submitted  for  suggestions  last  year,  and  a  record  of  phenomenal  interest  and 
discussion  in  all  parts  of  the  United  States. 

It  may  seem  unnecessary  to  describe  to  you  at  length  a  report  which  has  received 
such  remarkable  distribution  and  which  has  been  once  presented  in  tentative  form  by 
the  chairman  of  the  committee.  Nor  is  it  my  purpose  more  than  to  lay  before  you  the 
salient  points  which  may  form  a  certain  basis  for  discussion,  points  which  have  been 
suggested  largely  by  the  widespread  discussion  of  the  report  thruout  the  country.  As  an 
interesting  sidelight  upon  the  unusual  interest  manifested  in  this  document,  I  may  say 
that  the  committee  has  had  the  greatest  difi&culty  in  securing  a  sufficient  number  of  copies 
for  use  in  this  meeting,  since  the  edition  of  five  thousand  copies  authorized  this  year  is 
now  completely  exhausted.  Of  this  large  number,  nearly  three  thousand  have  been 
sent  out  by  the  commissioner  of  education  in  Washington,  each  copy  to  an  individual 
in  response  to  his  own  direct  request,  in  addition  to  the  two  thousand  sent  out  by  the 
committee  and  the  publication  of  the  report  by  School  Science  and  Mathematics. 

This  demand  is  important  because  it  demonstrates  the  real  need  of  such  a  report, 
and  augurs  its  future  influence  and  importance.  It  leads  naturally  to  a  statement  of 
the  effect  which  we  hope  the  report  will  have  upon  the  country.  It  is  not  expected  or 
desired  that  the  report  should  be  followed  slavishly  by  anyone,  or  that  what  the  com- 
mittee has  set  down  should  be  regarded  as  final  in  its  details.  It  is  rather  the  general 
spirit  of  its  contents  which  we  hope  will  receive  your  approval  and  which  we  think  will 
form  a  basis  for  further  thought  and  discussion  by  the  individual  teacher  and  by  societies 
thruout  the  country.  It  is  not  the  intention  that  anyone  whatever  should  be  bound 
by  any  word  or  phrase;  rather  it  is  hoped  that  the  moral  effect  of  the  report,  backed  by 
the  compelUng  force  of  its  reasonableness  and  by  the  general  approval  of  thinking  teachers, 
will  carry  weight  with  those  who  think  and  debate  on  questions  which  it  touches.  As  a 
stimulus  to  thought,  as  a  basis  for  discussion,  as  a  suggestion  of  one  reasonably  consistent 
plan  approved  by  many  men  of  note,  it  cannot  fail  to  have  a  vitalizing  influence  upon  the 
teaching  of  geometry  thruout  the  entire  country.  To  this  end,  the  committee  has  stated 
repeatedly  that  details  are  not  supposed  to  be  binding  or  of  great  importance,  that 


48  NATIONAL  EDUCATION  ASSOCIATION 

individual  opinion  be  accorded  the  widest  scope  and  the  greatest  respect,  that  the 
arbitrary  power  of  examining  bodies  be  curbed,  and  that  schools  be  free  to  work  out 
their  own  plans,  using  whatever  of  this  report  commends  itself  to  their  intelligent  thought. 
Such  freedom  is  of  the  utmost  importance;  to  imagine  that  this  report  in  any  sense  limits 
the  freest  possible  individual  action  is  completely  to  reverse  the  progressive  tendency 
which  was  the  keynote  of  the  committee's  action;  but  it  is  equally  true  that  individual 
freedom  needs  and  demands  some  organ  for  the  expression  of  a  general  consensus  of 
opinion,  some  expression  of  the  views  of  leaders,  some  basis  other  than  the  existing 
arbitrary  demands  of  examining  authorities  upon  which  intelligent  plans  may  be  built. 
It  is  the  purpose  of  this  committee  to  present  to  you  such  a  basis,  freed  from  all  other 
considerations  than  the  best  teaching  of  geometry  to  young  students.  Your  acceptance 
of  it  will  bind  you  and  other  teachers  only  in  so  far  as  the  strength  of  its  conclusions 
is  convincing  and  unanswerable  to  your  minds;  it  will  force  the  acceptance  of  its  gen- 
eral standards  and  of  the  principle  of  freedom  of  teaching  by  all  examining  bodies;  it 
will  lend  the  great  power  of  your  moral  support  to  the  further  dissemination  of  the  ideas 
it  contains;  it  will  make  for  progressive  discussion  and  untrammeled  advancement  in  the 
teaching  of  this  important  subject. 

The  report  is  not  a  mere  list  of  theorems.  It  touches  upon  every  phase  of  the  teach- 
ing of  geometry.  Thus,  one  of  the  important  portions,  placed  by  the  committee  at  its 
very  front,  is  an  excellent  resum6  of  the  history  of  the  teaching  of  geometry  in  this  country 
and  abroad,  prepared  largely  by  Professor  Cajori,  of  the  committee,  who  is  well  known  as 
an  authority  on  the  history  of  the  subject.  These  historical  notes  bear  vitally  on  the 
teaching  of  the  present  day.  They  have  been  carefully  weighed  by  the  committee  in 
preparing  the  rest  of  the  report.  Indeed,  the  positive  effect  of  historical  knowledge  of 
the  teaching  of  the  past  is  so  evident  that  it  seems  desirable  rather  to  emphasize  the  fact 
that  the  changed  conditions  of  our  age  and  the  changes  both  in  the  organization  of  society 
and  in  the  accepted  theories  of  education  make  necessary  a  great  degree  of  caution  in 
accepting  without  question  a  position  sanctioned  by  historical  facts  alone,  and  render  it 
necessary  for  us  today  to  review  without  prejudice  experiments  of  the  past,  even  tho 
those  experiments  may  have  clearly  ended  in  failure  in  their  own  day.  Thus  an  attempt 
to  render  the  teaching  of  geometry  more  practical  in  the  Middle  Ages,  tho  unsuccessful, 
does  not  discourage  the  renewal  of  the  same  effort  now. 

The  attitude  of  the  report  on  the  question  of  logical  treatment  is  next  emphasized 
(p.  3).  SuflSce  it  here  to  say  that  the  committee  has  retained  the  logical  form  of  discus- 
sion, at  least  in  so  far  as  logic  means  correct  reasoning.  On  the  other  hand,  the  extreme 
formalism  too  often  arbitrarily  attached  to  logic  is  distinctly  abandoned,  and  the  insistence 
is  placed  rather  upon  the  students'  understanding  of  the  reasoning  than  upon  the  satis- 
faction of  ideal  standards  of  pure  formalism.  The  psychological  principles  which  under- 
lie the  learning  process  are  everywhere  recognized,  so  that  the  student  is  to  be  gradually 
led  into  the  realm  of  logical  proof  by  easy  stages,  rather  than  thrown  into  it  in  connection 
with  theorems  that  seem  to  him  self-evident. 

Axioms  and  the  axiomatic  treatment  of  the  whole  subject  are  retained,  but  axioms 
which  have  no  sound  psychological  basis — such  as  the  so-called  "axioms  of  order" — are 
excluded.  Definitions  are  called  for  in  a  form  which  will  support  sound  reasoning,  but 
an  excess  of  formal  terms,  such  as  scholium,  and  the  introduction  of  new  terms,  are  decried. 

Informal  proofs  of  many  propositions  are  sanctioned,  at  least  in  many  cases  in  which 
the  theorems  are  almost  self-evident.  While  the  committee  has  limited  itself  in  this  matter 
more  than  some  teachers  desire,  the  acceptance  of  the  principle  of  informal  proof  will 
prove  most  acceptable  to  a  great  majority  of  teachers.  It  is  to  be  understood  that 
propositions  of  considerable  difl&culty  and  propositions  of  especial  importance  are  not  to 
receive  such  informal  treatment. 

The  topic  of  limits  and  incommensurable  cases  receives  special  attention,  and  it  is 
hoped  that  the  position  taken  will  commend  itself  as  reasonable  and  as  unprejudiced. 


COMMENTS  ON  THE  REPORT  49 

While  the  committee  was  in  thoro  agreement  regarding  these  topics,  it  was  felt  that  a 
statement  which  avoided  any  extreme  position  might  best  avoid  any  appearance  of 
dictation. 

The  statement  regarding  the  motives  for  the  study  of  geometry  (p.  12)  deserves 
notice.  It  should  be  remarked  that  the  old  appeal  to  discipline  as  a  motive  has  here 
yielded  to  other  more  convincing  claims. 

Special  courses  for  special  classes  of  students  received  more  consideration  than  the 
report  would  indicate  on  its  face.  The  final  statement  (p.  14)  expresses  the  conviction 
of  the  committee  that  these  special  courses  are  too  varied  to  permit  of  detailed  suggestions 
to  fit  each  case  which  may  arise.  It  is  the  hope  that  the  arrangement  for  emphasis  of 
more  important  theorems  will  itself  solve  this  difficulty,  in  offering  a  sane  basis  for  selection, 
when  a  complete  course  is  not  possible  or  desirable. 

In  making  suggestions  for  geometry  in  the  more  elementary  grades,  the  committee 
outlined  possible  work  of  a  geometric  nature  suited  to  the  needs  of  very  young  children. 
Perhaps  the  greatest  insistence  needed  here  is  that  a  clear  and  emphatic  distinction  be 
drawn  between  geometry  on  the  one  hand  and  logic  (i.e.,  formal  logic)  on  the  other  hand. 
Too  often  these  are  confused  thru  their  traditional  condition;  and  geometry  is  supposed 
to  mean  logical  deduction.  It  will  perhaps  be  clearest  to  say  that  the  committee  was 
unanimous  and  emphatic  in  opposing  any  deductive  logic  whatever  in  the  geometry  to 
be  taught  in  elementary  (graded)  schools. 

Very  great  attention  is  paid  by  the  committee  to  the  important  question  of  exercises. 
A  thoro  reading  of  this  portion  of  the  report  is  recommended,  since  it  cannot  be  reproduced 
in  brief.  Concrete  problems  of  a  more  practical  character  are  urged,  and  a  large  number 
of  illustrative  exercises  are  stated.  The  sources  of  such  problems  are  discussed,  and  the 
reasons  for  their  introduction  are  stated  (p.  19  and  p.  20). 

Both  here,  and  in  the  syllabus  of  theorems,  attention  is  called  to  the  use  of  algebraic 
methods  of  proof,  which  will  tend  to  insure  a  higher  degree  of  correlation  of  these  subjects, 
and  which  will  afford  better  and  simpler  means  of  proof  of  many  theorems  and  exercises. 

Before  an  attempt  is  made  to  read — or  even  scan — the  hsts  of  theorems,  it  is  highly 
desirable,  and  indeed  absolutely  necessary  to  a  correct  understanding,  to  read  the  preface 
to  the  syllabus,  pp.  30-33.  I  may  state  a  few  of  the  fundamental  points  which  might 
lead  to  utter  misunderstanding  if  overlooked. 

a)  The  theorems  are  here  stated  for  the  teacher — not  in  the  form  most  desirable 
for  students.  For  example,  many  are  written  in  abbreviated  literal  notations.  Many 
are  combined  into  triple  or  multiple  statements.  The  committee  distinctly  recommends 
that  these  be  re-worded  for  students. 

b)  Distinctions  in  importance  and  in  corresponding  emphasis  are  indicated  by  the 
use  of  four  grades  of  type.  The  theorems  in  black-face  type  are  most  important,  those 
in  italics  next,  those  in  ordinary  type  next,  those  in  fine  type  are  quite  subordinate  and 
verge  into  those  which  are  not  expHcitly  included  in  the  list.  It  will  be  a  great  mistake, 
however,  to  suppose  that  the  committee  regards  this  classification  as  absolutely  final 
or  binding.  Some  theorems  may  be  a  Httle  more,  some  less,  strongly  emphasized  than 
would  follow  from  the  types  here  used,  at  the  discretion  of  the  teacher  or  school.  Again, 
it  is  the  principle,  and  not  the  details,  for  which  the  committee  would  strongly  contend. 

c)  No  logical  order  is  intended.  Any  apparent  indication  of  a  logical  order  is 
unintentional. 

d)  The  principle  upon  which  the  theorems  are  grouped  is  that  of  their  fundamental 
connection  and  dependence,  not  only  logically,  but  also  in  their  practical  apphcations 
in  mathematics  and  elsewhere.  Here  again,  any  changes  whatever  desired  by  anyone 
will  be  sanctioned  by  the  committee.  Again,  it  is  only  the  principle  for  which  we  contend: 
that  attention  be  paid  in  grouping  theorems  to  the  fundamental  relationships  between  the 
theorems  placed  in  the  same  group,  so  that  a  possible  basis  for  a  human  grasp  of  the 
entire  field  is  furnished.  Such  a  grouping,  as  opposed  to  an  incoherent  arrangement 
which  depends  only  on  accidental  superficial  similarity  in  topic,  is  thought  to  be  quite 
vital. 

e)  Trigonometric  ratios  are  introduced  in  their  simplest  form,  as  they  actually 
present  themselves  in  geometric  figures. 


50  NATIONAL  EDUCATION  ASSOCIATION 


A  full  discussion  of  the  syllabus,  and  in  fact,  of  the  report  as  a  whole,  is  expected 
and  earnestly  requested  by  the  committee.  It  would  be  out  of  place  for  me  to  enter  upon 
a  time-consuming  exposition  of  the  details  of  the  syllabus,  which  lies  before  you  in  printed 
form.  I  may  say  that  the  statement  on  pp.  45-46  gives  a  clear  idea  of  the  view  of  the 
committee  concerning  the  treatment  of  the  several  grades  of  theorems. 

If,  as  a  result  of  your  discussion  here,  it  appears  that  there  exists  a  fair  consensus 
of  opinion  favoring  the  general  principles  set  forth  in  the  report,  we  of  the  committee 
will  feel  that  our  labor  has  not  been  in  vain.  It  is  neither  expected  nor  desired  that  all 
should  approve  each  detail.  Such  minute  agreement  is  beyond  human  belief  or  power. 
Even  the  committee  does  not  wish  to  be  understood  as  standing  unanimously  for  each 
and  every  minor  phrase,  tho  there  is  no  statement  which  has  not  the  unqualified  approval 
of  quite  a  majority,  and  there  is  nothing  of  moment  on  which  the  entire  committee  does 
not  agree  in  the  matter  of  broad  principle.  It  is  on  just  such  a  basis  that  we  hope  for 
your  approval  and  support.  With  it,  we  can  be  assured  of  a  hearing  by  every  teacher 
interested  in  geometry,  we  can  be  assured  that  the  earnest  efforts  of  this  committee  will 
not  go  unheeded,  but  that  they  will  be  given  consideration  wherever  the  subject  of  the 
teaching  of  geometry  is  discussed.  Precisely  this  we  do  desire — a  hearing  by  all  interested 
in  these  topics,  consideration  of  what  we  here  present  by  all  who  may  plan  courses  or 
discuss  the  planning  of  them.  To  desire  more  than  a  mere  hearing  not  only  would  violate 
the  freedom  which  the  committee  regards  as  essential  to  the  best  growth  of  the  schools 
of  the  country,  but  it  would  also  be  a  confession  of  our  lack  of  faith  in  our  own  conclusions. 
To  deny  that  hearing  would  be  just  as  great  a  blow  at  the  freedom  of  the  schools,  and 
just  as  great  a  confession  of  weakness  by  those  who  deny  it.  We  ask,  therefore,  your 
more  careful  consideration  and  discussion,  to  the  end  that  you  decide  whether  you  wish 
to  lend  the  moral  support  of  this  body  to  the  further  propagation  of  these  ideas  and  to 
the  stimulation  of  continued  thought  and  discussion  of  the  teaching  of  geometry. 


DISCUSSION 


Henry  W.  Stager,  head  of  the  Department  of  Mathematics,  Fresno  Junior  College, 
Fresno,  Cal. — It  needs  but  a  careful  reading  of  the  report  of  our  committee  to  realize 
its  marked  excellence.  In  recent  years  several  associations  of  mathematics  teachers  have 
published  syllabi  on  geometry.  While  these  attempts  have  been  valuable  steps  in  the 
progress  toward  a  rational  standardization  of  the  subject,  the  present  syllabus  is  the 
first  to  have  awakened  a  real  national  interest,  a  very  essential  factor  in  any  advance 
movement.  The  personnel  of  the  committee,  and  the  vast  amount  of  work  they  have 
expended  on  the  report,  remarkable  equally  for  its  great  liberality  and  its  comprehensive 
nature,  place  this  syllabus  in  a  position  by  itself. 

Whatever  other  claims  there  may  be  for  the  study  of  geometry — and  there  are  many — 
the  chief  purpose  of  this  course  is  to  develop  sound  and  accurate  thinking.  That  this 
quality  is  woefully  lacking  in  our  young  people  needs  no  reiteration  in  this  gathering. 
Aside  from  giving  the  students  a  knowledge  of  the  fundamental  truths  of  elementary 
geometry  and  arousing  in  them  a  genuine  liking  for  the  subject,  the  aim  of  the  teacher 
should  be  to  select  and  present  the  subject-matter  so  as  to  bring  about  the  highest  devel- 
opment of  their  thinking  power.  With  this  view  the  attitude  of  the  committee  strongly 
accords,  and  their  report  with  its  well-rounded  syllabus  keeps  the  three  chief  objects  con- 
stantly in  view,  and  gives  to  each  its  proper  emphasis. 

The  feature  of  the  report  which  appeals  most  strongly  to  me  is  the  extreme  broad- 
mindedness  of  the  committee.  Thruout  they  have  exercised  the  greatest  care  not  to 
appear  dogmatic  in  their  attitude.  The  value  of  this  fair-minded  attempt  to  get  at  the 
fundamental  truths  of  geometry  in  the  light  both  of  its  historic  character  and  of  modern 
needs  cannot  be  overestimated.  Such  an  attitude  demands  our  highest  commendation 
and  augurs  well  for  future  progress  in  the  teaching  of  geometry.     Whatever  the  view- 


COMMENTS  ON  THE  REPORT  51 

point  of  the  individual  teacher,  the  report  can  be  used  to  advantage.  For  instance,  many 
of  us  are  urging  the  teaching  of  mathematics  as  opposed  to  the  teaching  of  algebra,  geome- 
try, and  trigonometry  as  separate  subjects.  The  committee,  and  very  properly  I  think, 
does  not  consider  a  discussion  of  this  matter  within  the  province  of  this  report.  The 
syllabus  is  appHcable  equally  well  to  either  method.  Geometry  is  geometry,  whether 
taught  separately  or  in  correlation  with  other  subjects.  In  this  connection,  however, 
I  believe  that  the  committee  without  in  any  way  weakening  their  position  might  recom- 
mend in  their  statement  regarding  entrance  examinations  that  examiners  give  full  credit 
to  work  done  under  the  correlation  method. 

The  report  opens  with  an  excellent  historical  introduction  and  a  very  valuable 
bibliography  by  Professor  Florian  Cajori.  The  attitude  of  the  committee  (on  p.  3)  is 
so  admirably  expressed  that  it  needs  no  further  comment  here,  other  than  again  to  point 
out  its  breadth  of  view. 

The  suggestion  that  the  elements  of  trigonometry  be  introduced  in  connection  with 
similar  triangles  deserves  hearty  approval.  The  trigonometric  ratios  afford  not  only  the 
means  of  enlivening  an  ofttimes  very  trying  section  of  the  course,  but  also  the  oppor- 
tunity for  presenting  some  of  the  universal  applications  of  geometry.  I  have  noted  with 
much  pleasure  in  various  high  schools  the  effect  a  few  lessons  in  trigonometry  have  had 
on  the  classes.  A  new  interest  was  at  once  manifest  and  the  remaining  work  of  the  course 
was  carried  on  with  renewed  vigor.  This  work  should  be  introduced  even  at  a  sacrifice, 
but  in  the  average  nine  and  a  half  or  ten  months'  course,  I  have  found  that  teachers  can 
usually  find  the  necessary  time  without  omitting  any  of  the  essentials. 

With  the  position  of  the  committee  on  definitions  I  must  take  issue.  I  heartily  agree 
with  the  general  principle  that  new  terms  should  be  introduced  and  accepted  meanings 
be  changed  only  when  the  new  has  a  decided  advantage  over  the  old  and  has  the  general 
sanction  of  the  mathematical  world  and  not  by  the  action  of  individual  teachers  and 
writers.  However,  it  seems  to  me  that  our  committee  is  losing  a  valuable  opportunity 
to  standardize  to  a  large  degree  the  terms  and  symbols  of  geometry,  a  need  which  every 
teacher  recognizes.  The  initiative  in  this  movement  cannot  well  be  undertaken  by  any 
body  with  greater  authority  and  backing  than  this  committee.  And  this  action  must 
be  indefinitely  delayed  if  not  undertaken  now;  for  when  this  report  is  finally  accepted  it 
will  probably  be  many  years  before  a  similar  task  is  again  undertaken.  I  would  urge 
that  they  present  a  reasonably  complete  list  of  those  terms  and  symbols  essential  to  an 
elementary  course,  classifying  them  as  far  as  possible  in  much  the  same  ways  as  the 
theorems  have  been  classified.  Naturally  such  a  list  would  not  be  exhaustive,  but  it 
would  undoubtedly  tend  toward  a  greater  uniformity.  The  present  plan  fails  to  meet 
the  real  needs  of  the  situation.  For  instance,  it  is  suggested  that  "  mixed  line  "  be  dropped 
because  it  is  an  antiquated  term  of  no  value  in  elementary  geometry.  We  will  all  agree 
to  the  action,  but  what  similar  terms  are  suggested  for  dropping  by  this  term  ?  In  regard 
to  the  symbol  for  congruence,  I  would  like  to  urge  the  symbol  ^ .  As  far  as  possible 
symbols  should  be  self-explanatory,  i.e.,  they  should  suggest  readily  on  sight  the  operation 
or  form  they  stand  for.  Congruence,  as  used  in  geometry,  implies  equality  in  every 
respect;  i.e.,  in  both  size  and  shape,  as  opposed  to  equivalence,  equality  in  size  or  area; 
and  similarity,  equality  in  shape.  The  symbol  suggested,  ^ ,  meets  the  need  of  indicating 
this  double  equality  at  once,  the  upper  part  indicating  similarity  and  the  lower  part 
equality  of  size.  The  symbol  of  identity  suggested  by  the  committee  does  not  afford  this 
advantage  and  presents  an  old  symbol  to  young  minds  with  an  interpretation  somewhat 
different  from  that  to  which  they  have  been  accustomed.  This  same  symbol  has  a  use  in 
geometry,  which  can  best  be  shown  by  an  illustration.  Consider  the  congruence  of  the 
two  triangles  formed  by  the  median  AM  oi  the  isosceles  triangle  BAC.  In  selecting  the 
three  equal  parts,  it  is  of  value,  tho  not  essential,  to  say  ^M=^M. 

The  attitude  of  the  committee  on  informal  proofs  and  the  treatment  of  limits  and 


52  NATIONAL  EDUCATION  ASSOCIATION 

incommensurables  will  commend  itself  very  strongly  to  the  teacher  of  the  average  class. 
Care  must  be  exercised  not  to  use  the  informal  proof  too  readily  and  to  apply  the  method 
only  when  the  logical  proof  is  beyond  the  student's  knowledge  at  the  time  or  would  not 
appeal  to  him  as  a  rigorous  proof.  The  work  on  limits  and  incommensurables  usually 
should  certainly  be  confined  to  explanation  and  illustration  of  the  underlying  principles — 
and  largely  illustration.  An  attempt  to  prove  theorems  only  leads  to  hopeless  confusion 
and  memory  work,  which  is  greatly  to  be  deplored,  and  is  a  far  greater  injury  to  the  real 
progress  of  the  student  than  the  failure  to  get  a  few  propositions  of  doubtful  value  for 
an  elementary  course. 

Section  C  takes  up  the  consideration  of  special  courses.  In  this  connection  careful 
attention  must  be  given  to  the  recommendation  for  introductory  courses  in  the  lower 
grades.  While  the  needs  of  the  student  who  concludes  his  school  life  with  the  grammar 
school  are  manifest,  it  is  very  easy  to  carry  the  so-called  inventional  methods  too  far  and 
so  destroy  the  real  value  of  the  secondary  course  in  geometry.  Geometrical  drawing  is 
also  recommended  for  the  grades.  If  limited  to  the  use  of  the  ruler  and  compass  in  the 
accurate  drawing  of  simple  geometrical  forms,  some  advantage  may  result.  Too  often, 
however,  the  student  acquires  a  mere  mechanical  knowledge,  which  necessitates  the 
greatest  effort  on  the  part  of  the  teacher  of  geometry  to  direct  into  the  lines  of  logical 
construction. 

Special  courses  for  special  classes  of  students,  as,  for  instance,  those  in  the  manual 
arts  or  in  agriculture,  seem  hardly  warranted.  Such  courses  are  very  apt  to  defeat  the 
very  ends  claimed  for  them,  and  certainly  the  chief  aim  of  geometry  will  not  be  met  by 
any  lowering  of  the  requirements  to  meet  the  purely  utilitarian  viewpoint. 

Section  D  presents  the  question  of  exercises  and  problems.  The  tendency  for  many 
years  to  constantly  increase  the  number  of  exercises  has  naturally  led  to  the  introduction 
of  many  exercises  of  an  abstract  character,  most  of  which  serve  merely  as  mathematical 
puzzles  for  the  brighter  students.  Undoubtedly  there  should  be  a  large  number  of  con- 
crete applications  of  the  principles  of  each  theorem  (most  of  them  of  not  too  great  diffi- 
culty) in  close  connection  with  the  propositions  involved.  These  applications  should 
be  as  practical  as  possible  and  in  sufl5cient  number  for  the  teacher  to  select  different  sets 
in  successive  years  or  for  different  classes.  All  exercises  should  be  graded  with  the  great- 
est care.  A  carefully  selected  list,  including  some  of  greater  difficulty,  at  the  conclusion 
of  a  topic  would  afford  the  better  pupils  ample  opportunity  for  the  exercise  of  their  abiUty 
and  for  the  satisfaction  received  from  accomplishing  a  difficult  task.  In  my  opinion  the 
elementary  course  presents  no  opportunity  for  the  abstract  mathematical  puzzle,  unless 
in  a  chapter  especially  devoted  to  that  class  of  exercises.  On  the  other  hand,  there  is  the 
danger  of  reaching  out  to  the  other  extreme  of  the  so-called  practical  problems.  After 
reading  thru  many  of  the  texts  on  applied  mathematics  for  secondary  schools,  one  finds 
it  extremely  difficult  to  select  even  a  few  problems  which  will  remotely  interest  the  aver- 
age student.  Such  pupils  are  not  interested  in  the  details  of  architecture,  machine 
design,  railroad  curves,  and  the  like.  Their  problems  must  be  of  a  more  fundamental 
type,  intimately  associated  with  the  more  or  less  common  experiences  of  all.  In  this  con- 
nection, it  will  be  well  to  recall  the  value  of  the  trigonometric  ratios  already  suggested. 
Most  students  will  find  pleasure  in  a  method  to  obtain  the  height  of  the  flag-pole  on  the 
school  building,  or  the  distance  across  the  local  river,  or  out  to  an  island  in  the  bay,  or 
the  height  of  a  distant  mountain.  Again,  in  country  districts,  the  application  of  areas 
and  perimeters  of  triangles  are  numerous.  But  what  interest  does  the  average  student 
find  in  the  confused  lines  of  church  windows,  or  the  difficult  construction  of  the  railroad 
curve  or  the  three-centered  arch  ?  For  instance,  of  the  problems  suggested  as  types  by  the 
committee,  problem  9,  the  construction  of  the  three-centered  arch,  and  problem  11,  the 
church  window,  both  on  p.  23,  fail  in  a  large  measure  to  meet  their  own  requirements  and 
would  prove  of  slight  interest  to  the  average  student.     The  figure  of  the  former  problem 


COMMENTS  ON.  THE  HESQ^'^^  ;•.  ;.  j  ;    .^  53 


I 


is  misleading  and  the  statement  of  the  problem  itself  is  ambiguous.  In  both  cases  the 
scrlution  is  not  at  all  evident,  and  even  when  it  is  indicated  as  in  the  figures,  the  proof 
needed  is  too  difficult.  As  far  as  possible,  such  problems  should  be  avoided  and  they 
certainly  should  not  be  given  as  types. 

A  valuable  source  for  exercises  is  found  in  problems  involving  loci.  With  their  many 
applications  they  afford  excellent  opportunity  for  introducing  the  important  concepts 
of  motion  and  functionahty.  The  committee's  suggestion  that  problems  be  stated  in 
both  forms,  "locus  of  points"  and  "locus  of  a  point,"  to  bring  out  clearly  two  important 
but  different  phases  of  a  locus,  will  meet  with  no  opposition.  It  applies,  in  principle,  to 
all  classes  of  exercises  and  propositions.  Wherever  possible  it  is  desirable  to  employ 
different  forms  of  statement  to  make  clear  the  fundamental  ideas  and  give  facility  in 
expression.  The  character  of  the  loci  problems  should  be  broadened  to  cover  a  large  class 
of  applications.  Almost  without  exception  our  textbooks  restrict  this  method  to  prob- 
lems involving  distance,  so  much  so  in  fact  that  students  come  to  define  locus  in  terms  of 
distance.  While  four  of  the  Ust  of  nine  illustrations  given  on  p.  28  do  not  involve  distance, 
I  should  be  glad  to  see  the  broader  applications  of  locus  emphasized  more  fully. 

When  the  committee  urges  the  introduction  into  geometry  of  algebraic  methods  and 
of  problems  involving  algebra  it  well  states  that  "the  interdependence  of  algebra  and 
geometry  is  a  matter  of  no  small  importance  both  historically  and  for  subsequent  mathe- 
matical work."  The  correlation  of  the  two  subjects  often  affords  a  decided  advantage 
over  the  use  of  either  alone,  and  surely  there  can  be  no  objection  when  the  proofs  are  clear 
and  rigorous.  Again,  results  which  are  obtained  with  difficulty  by  the  use  of  geometric 
methods  solely  may  often  be  readily  deduced  by  a  combination  of  both.  Generally  there 
is  the  additional  gain  of  a  new  insight  into  the  problem.  The  construction  of  the  exact 
square  roots  of  prime  numbers  by  the  use  of  the  Pythagorean  theorem  lends  a  new 
interest  to  the  algebraic  extraction  of  square  roots.  Algebraic  formulas  constructed 
geometrically  become  alive.  Where  time  permits,  and  sufficient  interest  can  usually 
make  the  necessary  time,  some  of  the  elementary  principles  of  straight  lines  and  circles 
treated  analytically  will  afford  additional  means  of  strengthening  the  correlation  idea. 

Section  E  takes  up  the  syllabus  proper.  The  committee  presents  its  lists  of  theorems, 
which  it  definitely  states  is  not  exhaustive,  in  21  groups,  15  for  plane  geometry  and  6  for 
solid  geometry.  Within  each  group  the  theorems  are  divided  into  4  classes  according 
to  importance.  Certain  theorems  of  each  group  are  printed  in  black-face  type.  These 
theorems  are  considered  fundamental  and  are  to  serve  as  the  foundation  stones  of  geome- 
try. Theorems  next  in  importance  are  printed  in  italics.  A  third  class  is  printed  in 
roman  type,  and  a  final  list  of  the  least  important  is  printed  in  small  type.  The  aver- 
age textbook  presents  about  150  theorems,  exclusive  of  corollaries  and  the  theorems  in 
proportion  which  are  not  considered  by  the  committee.  The  present  syllabus  consists  of 
105  theorems;  26  in  black-face  type,  18  in  italics,  34  in  roman,  and  27  in  small  type; 
the  last  of  these  the  committee  suggests  be  considered  largely  in  the  nature  of  very  impor- 
tant exercises,  so  that  the  number  of  required  theorems  is  reduced  to  78.  We  cannot  too 
strongly  commend  the  courage  of  the  committee  in  this  efficient  pruning.  The  modern 
tendency  has  been  to  increase  the  number  of  required  propositions  out  of  all  proportion 
to  their  usefulness  and  without  in  any  way  adding  to  the  rigor  of  the  course.  Naturally, 
to  those  who  contend  that  elementary  geometry  should  be  a  perfect  piece  of  absolutely 
rigorous  mathematical  logic,  the  action  of  the  committee  will  appear  fatal.  On  the  other 
hand  those  who  consider  this  course  as  the  means  of  giving  to  young  minds  a  first  con- 
ception of  logical  mathematical  processes  together  with  certain  fundamental  truths  which 
will  be  of  value  to  them  in  all  subsequent  work  find  decided  advance  in  the  new  syllabus. 
The  latter  seems  to  me  the  only  tenable  position.  A  few  things  learned  thoroly  give 
greater  power  and  more  knowledge  than  many  things  merely  skimmed  over.  The  lesser 
number  of  required  theorems  leaves  increased  opportunity  for  the  introduction  of  work 


54  ^A  TIONAL  EDUCA  TION  ASSOC! A  TION 

essentially  practical,  but  which  cannot  now  be  accomplished  because  of  lack  of  tii^e, 
such  as  the  simple  applications  of  the  trigonometric  ratios.  And  most  important  of  all, 
there  will  now  be  ample  time  for  original  exercises  and  the  development  of  logical  think- 
ing processes.  More  than  all  else  this  faculty  of  our  students  needs  cultivation  and  a 
large  number  of  not  too  difficult  exercises  arranged  in  close  connection  with  the  theorems 
involved  afiford  the  best  means  at  our  disposal  for  its  development. 

Concerning  the  omissions  necessary  to  make  this  reduction  it  would  be  impossible 
to  secure  unanimity  of  opinion.  The  committee  does  not  present  its  lists  as  final.  I  regret 
to  fimd  among  the  omissions  such  theorems  as  the  centroid  of  a  triangle,  the  Golden  Sec- 
tion, and  Hero's  formula  for  the  area  of  a  triangle.  All  of  these  play  an  actual  and 
valuable  part  in  the  subsequent  work  of  the  student.  The  first  and  third  are  very  useful, 
whether  he  goes  out  into  life  or  continues  his  mathematics,  while  the  historical  associa- 
tions of  the  Golden  Section  and  its  relation  to  the  construction  of  the  regular  pentagon 
and  the  five-pointed  star  should  give  it  a  place.  Theorems  of  this  type  should  not  be 
included  in  the  course  at  the  discretion  of  the  teacher.. 

I  should  also  be  glad  to  see  the  committee  place  more  emphasis  on  proportion.  Its 
treatment  should  be  carefully  outlined  and  not  be  left  to  the  work  in  algebra.  For  the 
examining  board  in  some  states,  California  for  instance,  does  not  include  ratio  and  pro- 
portion as  a  part  of  the  first  year's  course  in  algebra  and  consequently  the  subject  is 
usually  first  considered  in  geometry.  A  demand  for  a  purely  geometrical  treatment  of 
proportion  in  an  elementary  course  seems  hardly  to  be  warranted.  The  algebraic  treat- 
ment, correlated  to  the  ideas  of  line-segments  as  far  as  possible  after  the  manner  sug- 
gested for  other  geometrical  concepts  in  correlation  with  algebra,  would  best  meet  the 
needs  of  elementary  geometry. 

At  the  close  of  the  list  of  theorems,  the  committee  suggests  a  method  by  which  the 
syllabus  may  serve  as  a  criterion  on  the  part  of  the  college  examiner  and  the  teacher. 
The  basis  of  the  recommendation  is  emphasis,  depending  upon  the  importance  of  the  four 
types  of  theorems,  and  is  in  full  accord  with  the  spirit  of  the  report. 

From  the  standpoint  of  the  teachers  of  geometry,  esi>ecially  those  of  less  experience, 
the  arrangement  by  emphasis  will  prove  of  greatest  value.  This  offers  an  outlook  over 
the  entire  subject  gained  only  by  long  experience  and  an  extensive  knowledge  of  mathe- 
matics. If  properly  applied  in  future  texts,  it  will  serve  to  break  the  monotony  which 
now  discourages  the  student  because  it  affords  him  no  opportunity  of  evaluating  the  work 
he  is  doing.  This  emphasis,  with  a  topical  arrangement  so  far  as  it  is  consistent  with 
logical  order,  will  mean  new  life  in  the  classroom,  a  more  thoro  knowledge  of  geometry, 
and  increased  power  on  the  part  of  the  student. 


■■^^ 


A/\/ 


'°o^ 


^-5 


'^-.-V:^c;i>\> 


«!• 


